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MATH 3210-001 SPRING 2025 - Week 1 - Functions

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Summary

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This content is an introductory lecture for Math 3210, Foundations of Analysis I, focusing on the syllabus and foundational mathematical concepts. The course emphasizes developing mathematical maturity through rigorous proof-based learning, moving beyond intuition to formal understanding.

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okay uh let's

start uh welcome to the spring semester

um this is math

3210 first section foundations of

analysis one this is the first of a

Al um email is my last name

Usman at um

Utah

prepared okay um yeah I'm a postto here

at the University of Utah at the math department

department

really which is I guess that side um so

today is going to be syllabus day um

probably we're not going to talk about

anything uh in terms of math but still

here yeah I didn't write okay so the

syllabus is on canvas I'm not going to

hand out you know physical printouts

um on syus I didn't didn't write

anything about the office hours I just

said by

appointment uh based on my experience

that works sort of the better the best

um because otherwise especially at a

university like this it's hard to find a

time that works for most people or for

all all all people uh typically and then

people tend to not um uh come to office

hours and stuff like that so I'd rather

have you take the initiative to set up

appointments with me using email and all

that okay so so this is a proof-based

class and the emphasis will be will be

on developing mathematical

maturity so how many of you have heard

this phrase mathematical

maturity any guesses as the what that should

mean moving from like grade school just

professional

other maybe just like a deeper

understanding of why it works the way it

works yeah yeah I would say so too um I

mean the way I think about it is that

okay so yes proof making is is an

important part of doing math especially

pure math but a lot of you are not going

to be mathematicians as far as I'm

concerned based on your Majors

um you might well be but uh that's not

the expectation of this class as far as

I'm concerned mathematical maturity is

basically about the following you you

see a sort of um high level um math

thing that is definitely beyond your pay grade

grade

and uh you know instead of freaking out

uh because you don't understand the

symbols you might still freak out but at

the very least you should have being

able to develop some sort of like method

to tackle it this complicated thing that

you encounter that's sort of like

mathematical maturity I should also

mention that um most of the documents

that I'm going to put on canvas they

have hyperlinks and uh the canvas system

doesn't really do well with hyperlinks

but um if you were to download the PDF

files you can just click on them and so

if you were to click on mathematical

maturity it's going to take you a uh to

a a blog post by Terence sty one of the

uh greatest mathematicians of our time

uh perhaps of all time um of all times

and uh one of the things he says is that

um you know there's this pre-rigor error

this is sort of like the type of math

that you do in the standard calculus

classes um they justify that this should

be that way or that should be that way

based on intuition obviously this is

true and stuff and then you take a class

like this and that's really about

understanding the formalism and being

able to be rigorous a little bit but

then you don't stop there really uh um

you go beyond and that is what is called

a post rigorous level because on the one

hand yes it is good to be able to prove

things it's also important to be able to

uh approach things from an intuitive

point of view so in this class we're

going to learn how to prove things but

we also try to we also try to we also

will try to um switch over to this post-

rigorous stage a little

bit in particular on rigorous

argumentation rather than calculation of

the values of certain limits integrals

or derivatives I might still ask you to

compute things of course but uh because

there are certain complicated things

that you can't compute unless you

understand what is going on okay uh

we'll start with Elementary logic and

set theory discuss main methods of

documentation that are up to the modern

standards of mathematical rigor and then

discuss differential and integral

calculus of single variable real valued functions

functions

as adjacent uh topics we'll study the

convergence and Divergence of sequences

functions expectations as a student you

are expected to attend all lectures

which is four times a week complete 12

problem sets take two in-class midterm

exams and take a final

exam according to the university

regulations you should expect a workload

of uh approximately 8 to 12 hours per

week outside of lecture hours so that's

a lot of math math time uh you will have

some familiarity with many ideas and

Notions we'll discuss throughout the

semester however the focus on rigor can

be hard to get get accustomed to

especially if you're not familiar with

these things and it is like that if

you're taking this class you're not that

familiar um there's a certain culture

shock uh Associated to switching from

sort of calculate this to prove this

um and there's going to be times when

you see things uh you know statements

they're obviously true to you but then

I'm going to ask you to prove it and

then you're going to realize that oh

maybe proving it is not that

straightforward maybe it is

but um missing um missing lectures or

assignments can make it challenging to

catch up please stay engaged and reach

out if you encounter any disruptions

we're here to help but your hard work

and dedication will be essential to

getting back on track learning

objectives among the objectives in math

uh 30

to10 are that you learn how to use

logical quantifiers describe statements

and the operations of and or and not in

in terms of sets identify injective

surjective and bjective relations and

functions Parts mathematical proofs

distinguish levels of rigor in mathematical

mathematical

Pros right proofs in accordance with

modern standards of mathematical rigor

which is basically um the standards that

were established let's say early in the

20th century now with the Advent of AI

and other uh proof cheer systems and

other things um now

what a valid proof is is sort of like

going through some revision but for our

purposes that's not going to be sort of

uh with in our horizon or maybe it'll be

it'll be in our Horizon but we're not

going to do something on lean or

something like

that approach numbers from a bottom up

point of view and make use of the cus

axium and the archimedian

property use the Notions of supremum and

inum you see this is where sort of

things start to get a little complicated

so we have Max and Men of course but

then there are these Associated or

similar Concepts supremum instead of

Maximum and in instead of minimum they

sort of like pretty much the same but

not quite and uh somehow supremum and

inum are terms that are more

mathematically robust and we're going to

have to learn what they mean and learn

how to use them derive the consequences

of the Triangular inequality inequality

for numbers and functions use sequences

and series of numbers and functions

identify when a sequence or series

converges and in which sense so

typically in a calcul calus uh in a

standard calculus course um you're given

a series or whatever and they ask you

what is the what is the

sum now we're going to see we're going

to encounter a lot of Series in this

class that are going to be possibly hard

to evaluate but still instead of finding

the limit you can ask for something less

which is that whether or not it

converges okay and probably you've seen

some tests that tell you this series

converges and not in like we're going to

look into that as

well use the Notions of accumulation

points limit inferior and limit Superior

of a sequence of numbers this limit

inferior and limit Superior are related

to these weird Notions supremum and eum

that we're going to study before prove

and use the fundamental theorems

governing continuous functions including

the maximum principle and intermediate

value theorem how many of you remember

the inter intermediate value

okay which basically says just to remind

ourselves it's nothing crazy okay so if

you have a continuous function

function

okay that's a continuous function why is

it continuous because I was able to draw

its graph without leaving without um

taking the um the the marker off the

discontinuities intermediate value

theorem says that your function takes

any value in

between okay obviously that should be

true we're going to see why we're going

to prove it rigorously

um uh distinguish continuous functions

from uniformly continuous functions so

that's a different type of continuity

that in fact Humanity had to deal with

for a while U before all this stuff was

uh polished and uh understood properly

prove and use of fundamental theorem

theorems governing differentiable

functions including the chain rule

inverse function theorem mean value and

Ro theorems and Lal rule probably most

of these are familiar to at least

somewhat um you might be U you might

remember them from your earlier classes

Define the notion of remon integrability

now as far as I'm concerned when it

comes to

integration we're going to have the most

Divergence from what you learned

before um because somehow

defining the remon

it's it's not as straightforward as as

defining derivatives um

straightforwardly and uh in a schedule

to um you're going to

see that we're going to have like four

weeks for

integration and of course prove and use

the fundamental theorem of calculus this

is the main probably topic of this class

prove and use the fundamental theorems

governing REM integrable functions

including the change of variables

formula so the use substitution right

and integration by parts Define

Elementary functions like exponential

and logarithm and prove and use tailor

approximation okay I'm not going to go

through the schedule uh you can read it

on your

own um so I like to be as specific as

possible when it comes to the schedule

but it's tentative of course and there

are always like things that ends up

needing to be changed and all that stuff

so we'll see I'm going to try to stick

as much as possible to the schedule but we'll

we'll

see and like I said

uh uh maybe I should comment on it a

little bit so the

foundations part is

basically maybe two to three weeks so it

is deemphasized compared to other sort

of reincarnations of this class there

many instructor who taught this class

over the years and as I was designing

this uh syllabus um I sort of looked at

pretty much all of them and I sort of

like changed a chang changed it a little

bit so that it makes more sense uh for

me at least we're going to spend two

weeks on continuity and two weeks on

differential calculus then four weeks on

integral calculus it's not exactly four

it's three and a half but still and then

we're going to have three weeks on

series you see the more messy Parts

we're going to to spend more more time

okay grades

grades

um so 5%

attendance I'm going to take attendance

classes I

% problem sets we're going to have 12 of

them they're going to contribute 35% uh

30% sorry to the final grade and uh our

TA this uh

semester uh is a strong ta as well so

I'm happy about that

one uh

okay a starts at 90% And then you drop

5% for each

uh letter degradation so a minus is 85

okay textbook that we're going to use

typically is the main textbook the

university uh or the the one that the

dep Department uh suggests uh Joe

Taylor's uh foundations of analysis Joe

Taylor was a faculty here for a long

time I think um so it's the in-house

textbook um and you know in class I'm

going to try to stick as much as

possible to the textbook I might change

things a little bit but uh still and um

as supplements as supplements I would

suggest looking at Terence st's lecture

notes and uh lecture notes by this

Turkish mathematician oh by the way I'm

Turkish as well uh this Turkish

mathematician aliar and um it's kind of

a weird thing but

um I thought it would be nice to share

with you the source that I learned this

stuff from and uh presumably I mean I

learned this

stuff maybe 10 years ago or something

but you know over time

the way you learn things for the first

time sort of seeps into everything so it

might be instructive to have a look at

these notes as

well okay any questions so far

far

yes uh it's on the syllabus

yes any other questions yes are any of

the problems that's um dropped at the

end of the semester possibly yes yeah I

mean uh depending on the

performance yeah

I typically drop two to three of them but

yeah any other

questions what can we expect the problem

sets to look like okay the problem sets

uh are going to be like this I'm going

to be preparing them myself but what I'm

going to do is this I use chat GPT to

scrape all the exercises from the book

okay and then I change them a little bit

and then there are a couple other books

that I like I look at them and stuff

like that so uh

I mean I have there there are other

classes that I taught online and you can

have a look at them and see

see

uh yeah I like to I I spend a lot of

time uh preparing the problem sets in um

um and the class is sort of like I try

to coordinate the class very uh

um let's say strongly related to

relative to the problem sets and wi

words and likewise the midterm exams so

the midterm exams typically are going to

be uh mostly from the problem

sets um but there might be one or two

questions that are sort of

new it's going to be like that um yeah

over time you'll you'll have a sense I

questions okay uh lectures attendance at

all lectures is expected attendance at

random lectures will be taken and will

have an effect on the final grades this is

5% I mean I just want you to come to all

classes basically and

um of course you know if you're sick and

stuff like that you should just let me

know ahead of

time um the number of lectures when

attendance will be taken will also be

random a certain number of recorded

attendance days may be dropped at the

um on weekends when a midterm is

scheduled so the weeks 5 and 11 first

two classes will focus on topics as

usual whereas the class before the

midterm is reserved for review likewise

the two classes on the last week which

is the which is the uh which is week 15

will be reserved for

review and I'm recording the lectures as

you see and uh there's a YouTube

playlist and at the end of each week I'm

going to sort of edit them together and

there's going to be one video per week

and uh um I think based on experience

I've been doing this for a while now and

based on experience it is very helpful

um to students and for myself as well I

sort of forget what I talked about in

class and I can just look at the

lectures and all that stuff and another

thing I do is I annotate uh the video so

that you have sort of oh you know this

is an example this a theorem this is the

proof of the theorem and so

on so I hope uh you'll make use of this

um problem sets 12 problem sets will be

assigned throughout the semester

typically of on a weekly basis excluding

weeks with midterms so basically uh

we're going to have one problem set du

every Friday except the Fridays when we

have a midterm okay and then there's

also a problem set do the day the final

schedule which is another another Friday um

so at any given time you're only going

to have one thing to worry about either

you're preparing for a midterm or are

you're trying to complete a problem

set late submissions up to 24 hours

after the deadline will be accepted with

a 10% penalty uh submissions more than

24 24 hours late will not be accepted

unless you contact the course staff with

a valid excuse before the 24-hour ex

extension expires

a certain number of problem sets with

lowest scores may be dropped at the end

of the semester grading

criteria uh the problem sets will be

graded partly on completion and partly

on accuracy validity and presentation

documentation so typically this means

the following typically we're going to

have let's say five or six

problems and uh I'm not going to tell

you which one I will be grading

specifically and when I grade a problem

set problem from a problem set that's

going to be for completion and the rest

uh typically is going to be creaded by

RTA so you're not going to know which

one which problem from a problem set is

going to be graded for accuracy but um

so hopefully that'll sort of uh push you

to um write your

answers you know properly for for all

problems submission details for each

problem set you are required uh to sub

MIT a form through Google forms and

digital copies of your work through

grade scope detailed submission

instructions will be provided in each

problem set specification so this

business about the Google forums that

has it has two functions uh one function

is that I ask how you're doing and stuff

so I'm going to ask things like okay how

long uh did it take you to finish this

problem set and I'm going to aggregate

aggregate all that data and based on

that we're going to sort of make

adjustments maybe I'm sort of like

asking too hard questions or um maybe

it's too easy and stuff like that and

the second of all is that I'm asking I'm

going to ask things like oh you know I

did this problem set by myself and I

um I mean of course you can lie but uh

it's not a you're not supposed to lie

um conflicting instructions and

deadlines the submission deadlines and

methods Det detailed in the problem set

specifications I.E the problem F PDF

files embedded in the associated

assignment web pages on canvas uh canvas

and grade scope May differ always adhere

to the deadline and instructions in the

specification document as these will be

uh the accurate guidelines so right now

on canvas if you look uh there are

assignments already there but they're

all empty as time goes by I'm going to

populate them okay so the for instance

problem say one is going to be due not

not this Friday but next Friday and I'm

going to release it um by this Friday

okay so you're going to have basically a

week to finish start and finish a problem

problem

set midterm exams two midterm exams uh

first one is on the Friday of week uh

five which is February 7 in class uh

covering topics from Week 1 to 5 Second

midterm on March 28

uh Friday of week 11 covering topics

from 6: to 11 and then the final there's

going to be a final exam during the

midterm exams the use of external

resources including colleagues books and

electronics will not be permitted your

exams will be digitized and uploaded to

great great scope by by the staff by me

and you will have access to view how

your exams were graded and likewise the

final exam is scheduled uh to be on

April 25th from 8 to 10 again here and

the final exam will be

cumulative again it's going to be close

books you cannot use anything except pen

far okay uh Math Center um especially

for this class I think Math Center is

pretty good um there's a bunch of uh

good people there that I know um

in addition to the office hours held by

the course staff the math Center is an

excellent resource For assistance with

your studies located in the basement

connecting the two math buildings so we

have two math buildings here over there

jwb which is where my office is at and

LCB and there's like a tunnel okay that

connects it to and there's like that's

that's where you find the math

Center the math Center offers free

dropin as well as online report

accessibility the University of Utah

seeks to provide equal access to its

programs services and activities for

people with disabilities if you will

need accommodations in this class

reasonable prior notice needs to be

given to the

CDA CDA will work with you and the staff

to make arrangements for accommodations

all written information in this course

can be available can be made available

in an alternative format with prior

notification to to the Center for

disability and

access inclusivity and safety Title 9

makes it clear that while violence and

harassment based on sex and gender which

includes sexual orientation and gender

identity expression is a civil rights

offense subject to the same kinds of

accountability and the same kinds of

support applied to offenses against

other protected categories such as race

national origin color religion age

status as a person with with a

disability veteran status or genetic

information if you or someone you know

has been harassed or assaulted you are

encouraged to report it to the title L

coordinator or the office of the dean of

students and there's a phone numbers and

stuff here if you're in that

situation for support and confidential

consultation contact the center for student

student

wellness um SSB 328

oh maybe they're moving actually never

mind uh to report to the police contact

the Department of Public Safety and just

you know call the cops and there are

further safety resources under safe

you and finally academic honesty you are

expected to adhere to University of Utah

policies regarding academic honesty this

means that ultimately all work you

submit must be your own created without

unauthorized assistance and all external

resources including generative tools so

this chat GPT type stuff and computer

algebra systems must be properly CED and

documented within your

submissions and you're going to do this

uh using the Google form that I was

mentioning um any student who engages in

academic dishonesty or who violat

violates the professional and ethical

standards may be subject to academic

sanctions as per the University of

Utah's uh student code generative Ai and

computer algebra systems in this course

you may use generative tools like chat

PT as well as computer algebra systems

like such as mathlab and probably

mathlab is not going to be too much help

but still might be useful under specific

guidelines these tools are permitted to

guide your understanding of problem sets

and topics when using them you must

submit submit logs of your interactions

as part of your problem set submissions

so for instance if you're using chat GPT

there's a function there that allows you

to share your uh

conversation um these tools can also be

used to verify calculations but remember

computations are generally expected to

be done manually unless explicitly stated

stated

otherwise so ultimately in this class um

we should expect to learn the type of

math that you should be able to do if

there was like a you know a civilization

collapse we don't have any electricity

or whatever or you know you're stranded

in a in an island and so for some reason

you want to do math and that's the type

of math we're going to

learn you are responsible for ensuring

the validity accuracy and relevance of

your submissions whether or not you use

these to tools while encouraged the use

of such technology is not required for

success in this class which means that

if you were to take this class you know

in the 1800 hundreds it's pretty much

all the same there might be some you

know new things but still

um the policy governing the use of such

technology is specific to math 30 to10

d001 this section uh other courses and

sections may have different policies and

we have a regret regret Clause it says

the following if you commit an

unreasonable unreasonable act but bring

it to the staff's attention within 48

Hours of the relevant submission

sanctions may be limited to that

submission only rather then leading to

further disciplinary action this Clause

will not be applied in the case of

repeated violations so let's say that

you're you know sleep deprived and stuff

like that and you're trying to finish

the problem set and you made the mistake

of leaving it to the last minute let's

say or last day perhaps and uh by you

know uh you have a lapse of judgment and

you sort of like search online and

there's you know stack exchange and

someone wrote the answer already and you

just copied and paste it pasted it and

like didn't say anything um after you

sort of like you know sleep a little bit

and all that stuff

uh if you let me know within 48 hours no

harm done that's what I'm saying

there okay and that's about it any

anything okay

um so we have time until

10:30 so we have about 20 more minutes

beginning so the title of week one I

called to be functions and really this

week we're going to uh study sort

of more abstract stuff a little bit sets

um logic stuff what is a function the

ration and so on and then starting from

um week two we're really going to uh

switch over to you know the reals the

real line which is sort of like the main

ambient space for well single variable

calculus right so foundations of an

foundations of analysis one really is we

should think of it as advanced calculus

um well Advanced single variable [Applause]

[Applause]

okay

so I don't want to spend too much time

on you know logic and sets and all that

stuff I'm just going to introduce some

notation today and you know throughout

this week and then hopefully you'll get

ACC custom to these things as the

semester progresses because the main

point of this class is not you

know understanding you

know logic itself or for instance the

construction of the reals it's like a

topic that the book spends quite a bit

of time but really to sort of connect

formalism and

calculus and so

symbols now these are not necessary

strictly speak strictly speaking but

certainly it helps quite a bit to

compress information and expression and

in this class typically what happens is

that you end up needing to write the

same thing over and over again so it's

going to sort of make it quite uh

simpler a lot simpler and so there's a

couple things one of them is what is

any okay and then there is something

and then there's a maybe we might use

unique okay these three uh terms we're

going to use quite often in this class

well for this I'm going to use a sort of

a okay for exists I'm going to use backward

backward

e see e e a a and then when you have it

the exists unique you just use uh

backwards e within uh um what do you call

it exclamation you're surprised somehow

that there's unique one okay I'm just

going to leave it at

this instead of sort of looking into

sort of random statements uh um I draw

their sort of move forward and then

start using these terms these symbols in

the context of

calculus okay and of course we have

other things um these have notations as

well but I'm not going to write them

and

or well there's X or which is uh well

I'll mention it later um what else not

implies so using these we're going to

connect a bunch of statements and so on

and we're going to have in our theorems

a bunch of like

Clauses we're going to have if this and

this and this then this and that and

that we're going to have stence like

this okay that's all we need for now sets

well the set is just a bunch of things

now even though in this class we're

really looking into rigorous things

we're not going to go all the way you

know in to se and stuff like that we're

just going to still leave it as you

know um as at the at the at the naive

level and because I draw there spend

more time on calculus

stuff um so a set is a bunch of things

there a

collection a

set is

things so one of the things one of the

important sets that we're going to use

of course is um the set of real

in fact let's write a bunch of number

systems so we have natural

numbers and so you wrote you write n for

natural and then you put a dash that is

what is called Blackboard

bold and really I'm not going to use

this notation I'm going to use some

other notation that I'm going to write

in a second there is

integers um so the natural start at one

I think uh 1 2 3 4 5 and so on integers

you have 0 1 2 3 4 5 and also - 1 -2 - 3

and so on so instead of n because you

know some people use n to denote

integers start at starting at zero some

other people start at one and so on I'm

just going to write instead Z

sub greater than or equal to one okay it

is more explicit you don't need to guess

is zero included is it not included in

whenever people start you know there's

this thing it's like some people say oh

you know is zero natural numbers like

you can just avoid it just say integer

I'm starting at one

whatever Q rationals so this this is the

ratio of two integers and then the next

thing that we're going to spend the most

time with is the

reals let me WR let me write the names

of these Naturals natural

now how do you describe these sets I

just wrote some symbols on the board and

named them and like how do you how do

there's a couple different ways of

writing down a set

um so for instance to write down the

first set over there set of natural

numbers one thing you can do is you can

just list them down and of course there

are infinitely many natural numbers so

how do you list them down you just put

ellipses and uh you just say okay this

is this is what I mean so you can write

this you open curly braces and then

you're going to close

it 1 2 3 four dot dot

dot okay now if you're a very um

formalist person you would say oh you

know this is a set with five elements

and you have four elements here and this

is the fifth elements but when we write

it like this we mean just the infinitely

many integers that we have over there

okay we're not going to be that strict

um integers well I can write it like

this 0+

on again if you're more you know

formally inclined inclined you would

want to write it like plus one and then

minus one with a comma and so on but

it's clear what I mean here okay and in

this class even though we're trying to

understand proofs and stuff like that

and learn how to write proofs I I don't

want to be a stickler about you know

being very formal

formal now one thing I should mention

here is that when you're writing these curly

curly

Braes um the order in which you write

things doesn't matter so this is an

unordered list if you want okay so I

could have just as well written Instead

This oh maybe I want to write 1 3 5 and

then it goes on and then 2 4 6 and then

it goes

on okay they are the same thing

thing

okay um next one

now at this point um here's a different

way of writing down a set you can do

this you pretty much always open curly

braces and then close curly braces but

instead of listing the elements what you

can do is you can use what is called a

set builder

notation which says The Following start

with open braces Clos braces put a bar

or maybe semicolon or colon I guess I

like to use bar to the left of the bar

you put the form of a typical element in

that set that you want in this case I

want to say something like this p over q

a typical element in a set of rationals

is going to be a ratio

right to the right of the bar you put

the conditions p and Q I need to say

something about them well p and Q are integers

integers

p and Q

integers

okay any issues with

this you allow Q to be zero right you

want to say something about Q not being zero

um why because you don't divide by zero

I mean there are other number systems

where you allow division by zero but

let's leave it at

that now describing R

um is going to be not that

straightforward uh we're going to have

to discuss how to go from Q to

R what it is is that somehow if you were

to think of all the rational numbers and

if you were to line up because given any

two rational numbers you can say oh this

one is smaller than the other one which

means that you can think of them as part

of a

Continuum well it turns out that if you

were to if you were to think of this

Continuum if you were to just stick the

rational numbers there's a bunch of

gaps because it turns out that there's a

bunch of numbers that are not rational

but still

real um so for now we'll just leave it

later and this was a problem that um

Humanity uh came across a long time AG

ago right because they were into

geometry and

stuff and

uh they looked at this right triangle

with two sides of equal length let's say

one and

then of course uh I'm sure a lot of you

know that this length should be < tk2

and then they said oh what is what is

how can I write root2 as p over

Q okay and then they realize that you

cannot do it you cannot write root2 as

as as a ratio of two

integers okay and then over time people

realize oh

maybe this rationals is kind of buggy

well at least to do some geometry and

stuff like that certainly but uh when it

comes to computers for instance it turns

out that real numbers is perhaps a

little bit obsolete because there's no

such thing as infinite resolution right

infinitely many um um digits after the

after the dots when you're trying to

write down a number at least for the

purposes of computation there's always a

certain cut off and stuff like that and

you don't really use it and then people

realize that even Beyond reals there are

more important things or more useful

things depending depending on the

context like the complex numbers or the

Quan and stuff like that um which we're

not going to get to

exercise

to try to come up with an argument let's say

uh for the statement that root2 is not a

rational number

for next time let's think about that let

me write it

now we're going to discuss later this

week what are sort of valid valid

methods of proof but let me just say for

now one thing you can use what is what

which means that you say okay say it is

number and here's a new symbol this

means elements right like I'm saying say

< tk2 is an element in uh in the

rationals I'm going to mention this

tomorrow as well but uh so there are you

integers Q not equal to Z such that <

Q your aim here is to keep pushing this

bizarre get a bizarre thing get a wrong

okay that's it for today thank you I'll

tomorrow okay let's start good morning

morning

um so yesterday we uh stopped at this

exercise that I gave you um which was uh

to prove that this number

root2 which pythagoreans you know

ancient Greek people were uh certain

existed I guess uh is not a rational

number that is to say it cannot be

written as the ratio of two

integers and I thought about it a little

bit and

um I want to uh leave it to you still um

perhaps until tomorrow or Friday where

we start talking about proof methods

more specific and I think I want to do

it in class as well uh because I think

it's a very instructive proof and it's

one of the classical proofs that you

sort of do when you start thinking about

how to prove

things um so we're going to continue

today on the foundations and stuff I

wanted to mention one thing which is

this Rin selection task or it's also

called the four card problem it's it's

something about logic um but it's also

part of like psychology and it was about

like um how people argue or think about

certain logical statements and stuff and

then once we talk about this a little

bit um I want to say a couple words

about how to approach statements in this

class so first of all how many of you

have seen have heard of

this it's a psychology experiment uh so

the experiment is the following

you're told that there's a deck of cards

these are supposed to be uh

cards such that on one side for each

card is a number is an integer

side and a color that is to say either

blue or red on the other side on the

okay this you're

given and now uh somehow they give you four

four

cards and you see the following one of

them is three the other one is eight

um the next one you see

blue so

red

okay this is the setup now the question

is the following you're given the

statements if

um yeah if a card has an even number on one

one

blue now they don't tell you if this is

correct or not okay and that's the

that's the task your task is to do this

your task is to um flip as few cards as

possible to verify or deny that this

statement is true

okay any questions about this so the

point is like you want to sort of like

test test the test the situation right

like maybe you want to flip this and

stuff but this is all you care about nothing

nothing

else the question is which cards should

you must you be uh flipping to verify or

deny the eight and the

blue eight and the blue any other

guesses um you would have to flip the

okay any

other you don't you don't have to flip

the blue

okay okay

so there's an intersection there that is

I think correct but um let's go step by

step do I need to flip

three I don't need to flip three because

the hypothesis is that if a card is has

an even number on one side so I don't

need to do anything about this I could

of of course but that's

um let me just write it

flip okay not three how about eight well

eight obviously is an even number

because you know a number being even

means it's divisible by two and if you

divide this by two you get four it is

divisible and so it has eight so I need

to flip it to verify so I need to see

maybe it is blue maybe it is red I have

eight like you were

suggesting now

blue this is where things get

complicated a little bit and this is

what is called what we're going to call

um Contra positive so this statement is

of the

form p a card having an even number only

on one

side let me not write it

implies some other statement Q which

means a card uh so Q stands for a card

has um blue on the other side turns out

that this is equivalent logically

equivalent and this is sort of the

notation that I use

so this is what we're going to call counter

positive which will be one of the main

methods of proof that we're going to

use now the reason why I I don't need to

flip this is that this statement doesn't

say anything about what happens if um on

one side um you have an odd number maybe

it is also the case that for some uh

cards for some odd numbers on the other

side you have blue I don't care but this

you see this is this

case um

Q was on one side it is blue not Q

here then stands for on one side it is

red this means that this statement is

equivalent to saying that if a

card has is blue sorry if a card is red

on one side on the opposite side it must be

well does this make

sense okay now um some of you are into

Eon and other things um and it turns out

that people did a bunch of different

variations of this experiment and it

turns out that if you sort of instead of

doing this abstract type of task if you

sort of make it more relevant to social

situation so there's a version where um

you're looking at so there's a bar and

there are people in there drinking

things and um you're trying to guess

you're trying to figure out

if um people are doing you know like

underage people are not drinking

alcoholic beverages let's say and you

say it's like I'm going to check the uh

age of the person and what what type of

drink they're drinking it turns out that

um more people are able to figure out

the logical uh you know what you what

you should be doing logically compared

to this abstract thing but of course in

our situation in this class we're going

to be doing abstract math stuff and so

that means that we're not going to have

that social cont um context to sort of

guide us there is some sort of intuition

of course that I hope you'll be able to

build in time but still

okay any questions about this I just

wanted to mention this um it's a nice logical

logical

thing and

this implication stuff is sort of one of

the things that we're going to be using

in this

class okay so how to approach statements

in math 3210 and in no particular order

I said over

there because

um when you see an exercise

or when you see a problem in a problem

set or a problem in or you know question

in an

exam I'm not going to necessarily tell

you that prove this okay what I'm going

to say is that I'm going to give you a

statement and your job first of all is

the is going to be to figure out if it is

correct and so I thought about this

program but I'm saying in no particular

order because you know the um cognitive

processes Associated to doing math is

somewhat complicated and it's beyond my

pay grade as well

well

so you want to answer you want to be

you want to figure out first of all if it is true or not now maybe it is

it is true or not now maybe it is actually true or maybe there's a typo

actually true or maybe there's a typo maybe there's a mistake maybe the author

maybe there's a mistake maybe the author of the statement that you're um the

of the statement that you're um the author of the statement that you're

author of the statement that you're reading just missed missed a detail or

reading just missed missed a detail or maybe

maybe they you know they weren't that that

they you know they weren't that that they weren't that careful about the

they weren't that careful about the details and stuff like that right so

details and stuff like that right so these are important things and uh if

these are important things and uh if possible you want to be sure if not you

possible you want to be sure if not you at least want to have a guess

at least want to have a guess okay possibly let's

say with some

probability okay

okay um so maybe you're going to encounter

um so maybe you're going to encounter certain statements that you cannot

certain statements that you cannot really distinguish at a certain time

really distinguish at a certain time whether or not they're true but at the

whether or not they're true but at the very least you want to have some guess

very least you want to have some guess you want to say okay 70% I am 70% sure

you want to say okay 70% I am 70% sure that this is correct there's also a

that this is correct there's also a theory of what it means to prove things

theory of what it means to prove things probabilistically especially in computer

probabilistically especially in computer science this is an important thing but

science this is an important thing but we're not going to get

we're not going to get there another thing you want to do is of

there another thing you want to do is of course build your case okay so you have

course build your case okay so you have a statement you see a statement

build a case when you see this abstract

case when you see this abstract statement you want to start thinking

statement you want to start thinking about examples

about examples non-examples counter examples non-

non-examples counter examples non- counter examples

counter examples okay

with examples non-examples

and I'm going to leave it to you to figure out what I mean by a non-example

figure out what I mean by a non-example counter example examples counter

examples further sometimes if maybe you you can't even do this for instance in

you can't even do this for instance in the case of proving that root2 is not an

the case of proving that root2 is not an integer not not a not a rational number

integer not not a not a rational number well what you can do is you can start

well what you can do is you can start start a proof okay we're going to learn

start a proof okay we're going to learn a couple different methods you can start

a couple different methods you can start a proof and you can try to sort of keep

a proof and you can try to sort of keep pushing and pushing and maybe you're

pushing and pushing and maybe you're going to get stuck somewhere and then

going to get stuck somewhere and then where you get stuck in the proof is

where you get stuck in the proof is going to tell you something about

going to tell you something about whether or not the statement is actually

whether or not the statement is actually true because ultimately if something is

true because ultimately if something is is false you cannot prove it but you can

is false you cannot prove it but you can still try to prove it and ultimately

still try to prove it and ultimately you're going to get stuck if it's false

you're going to get stuck if it's false really I mean you might get stuck of

really I mean you might get stuck of course you can always prove

uh uh something that is false or you can you can always uh convince yourself that

you can always uh convince yourself that you're proving something false

but that's up to you in a sense um start a proof and see if you get stuck

a proof and see if you get stuck somewhere

and this is going to happen quite often you're going to start to prove because

you're going to start to prove because you you s of sort of like you convince

you you s of sort of like you convince yourself that something is the case and

yourself that something is the case and you start to proof and then eventually

you start to proof and then eventually you're stuck in like you cannot see how

you're stuck in like you cannot see how we can go further and then something's

we can go further and then something's going to happen in your in your mind uh

going to happen in your in your mind uh quite possibly and you're going to say

quite possibly and you're going to say oh Sil in me clearly there's a counter

oh Sil in me clearly there's a counter example to this and here's a counter

example to this and here's a counter example

example um this is not the end four is what I

um this is not the end four is what I call uh wait three okay three is prove

call uh wait three okay three is prove prove everything

everything unless it was covered in class or in the textbook right those you

class or in the textbook right those you can take everything else you

can take everything else you prove um

prove um there's a item four so I want to leave

there's a item four so I want to leave it

it there um I'll write item four after

there um I'll write item four after deleting this actually it's sort of like

deleting this actually it's sort of like a Next Step

the textbook by U Joe Taylor no Bluffs

Taylor no Bluffs okay I will call your

Bluffs so what do I mean by this uh well statements like clearly this is the case

statements like clearly this is the case it's obvious it's trivial I mean it

it's obvious it's trivial I mean it might well be but uh you might be

might well be but uh you might be bluffing

bluffing so if you say something like that

so if you say something like that probably I'm going to be like oh I think

probably I'm going to be like oh I think you should prove

it uh clear

trivial in other less uh incendiary phrases it can be

proven you see this typically in calculus

calculus books uh General statement general idea

books uh General statement general idea is when in doubt prove it like you think

is when in doubt prove it like you think about oh there's a step should I prove

about oh there's a step should I prove it yeah you should prove

it so generally speaking you should always think of the following right like

always think of the following right like um there's this game it's like you start

um there's this game it's like you start with a

with a statement I don't know it's like when

statement I don't know it's like when it's called Uh um the so

it's called Uh um the so the a lake let's say starts um freezing

the a lake let's say starts um freezing from the top and then you say why and

from the top and then you say why and then you keep asking why and why and why

then you keep asking why and why and why and because it's like you know uh the

and because it's like you know uh the density of um ice is less and blah blah

density of um ice is less and blah blah blah

blah right you can always keep asking why and

right you can always keep asking why and in math to in principle you can always

in math to in principle you can always do this now whether or not you should be

do this now whether or not you should be doing it all the time that's a different

doing it all the time that's a different matter but um that's sort of like the

matter but um that's sort of like the general idea

yeah okay any questions so far now four is what I call logical

far now four is what I call logical perturbation now that you're done with

perturbation now that you're done with the

the statements it's never over okay math is

statements it's never over okay math is never

over so there's a distinction when it comes to games they say there's finite

comes to games they say there's finite games and infinite games and finite

games and infinite games and finite games are things like chess it's like a

games are things like chess it's like a you know you play and eventually there's

you know you play and eventually there's a there's a game over math is not like

a there's a game over math is not like that math is a game that you keep

that math is a game that you keep playing you play it to keep

playing you play it to keep playing um of course there are

playing um of course there are restrictions because this is a class and

restrictions because this is a class and all that stuff and we have only so much

all that stuff and we have only so much time and so much you also have so much

time and so much you also have so much time but

time but still logical perturbation is about the

still logical perturbation is about the following you take a statement let's say

following you take a statement let's say that you understood everything or maybe

that you understood everything or maybe you haven't you see in no particular

you haven't you see in no particular order but you start sort of changing the

order but you start sort of changing the statement a little bit a little bit you

statement a little bit a little bit you start perturbing the statement logically

start perturbing the statement logically speaking so for instance in the exercise

speaking so for instance in the exercise that root2 is not a rational number I

that root2 is not a rational number I can think about oh what is what is it

can think about oh what is what is it about two there that really sort of

about two there that really sort of helps helps me in the proof right maybe

helps helps me in the proof right maybe root three a similar argument could

root three a similar argument could possibly work there or

possibly work there or maybe I can I can use I can try for Root

maybe I can I can use I can try for Root 6 see that's sort of like generalization

6 see that's sort of like generalization so there are a couple different ways of

so there are a couple different ways of doing

this and let me write these down um oh yeah I'll write it

here so first of all you can try to weaken the statement

weaken the statement hypothesis so can you still prove the

hypothesis so can you still prove the same conclusion without assuming it as

same conclusion without assuming it as much so it can be more

hypothesis or you can try to be more Grey

Grey with the same hypothesis can you prove

with the same hypothesis can you prove something

stronger strengthen the

conclusions um what else well maybe um you can try assuming more so

maybe um you can try assuming more so maybe you Cann not prove this general

maybe you Cann not prove this general statement but maybe you can prove a uh

statement but maybe you can prove a uh more simple statement more uh

more simple statement more uh restrictive statement but maybe you can

restrictive statement but maybe you can get more

get more okay because if you sort of decrease if

okay because if you sort of decrease if you uh make your scope smaller typically

you uh make your scope smaller typically that means that you have more properties

that means that you have more properties which is which uh is reasonable um to

which is which uh is reasonable um to expect then that you can prove

expect then that you can prove more

more assuming more

assuming more can you prove more can you prove

more and another thing of course is can you come up with other proofs often

you come up with other proofs often there are multiple ways of doing the

there are multiple ways of doing the same thing and these multiple ways are

same thing and these multiple ways are not necessarily logically equivalent

not necessarily logically equivalent okay there are logical equivalences like

okay there are logical equivalences like this contraposition that I mentioned a

this contraposition that I mentioned a while ago but sometimes they're really

while ago but sometimes they're really uh mathematically or you know logic the

uh mathematically or you know logic the different ways of uh approaching

proofs maybe simpler proofs maybe more complicated but more concrete proofs and

complicated but more concrete proofs and stuff like

stuff like that okay this is sort of the general

that okay this is sort of the general scheme

scheme uh that you should use When approaching

uh that you should use When approaching um statements in this class any

um statements in this class any questions about

these okay um so we were talking about sets now that this is done let me give

sets now that this is done let me give another class of sets that we're going

another class of sets that we're going to be using quite a bit uh which are

to be using quite a bit uh which are what are called

what are called intervals these are subsets um of the

intervals these are subsets um of the real line

so let me also introduce these two uh symbols one of them I started using

symbols one of them I started using yesterday I think so

um this is going to stand for lowercase a is an element of uppercase a so here

a is an element of uppercase a so here lowercase a is a point or an object

lowercase a is a point or an object let's say and uppercase a is a

let's say and uppercase a is a collection is a is a

set okay another thing is uh the following when I write like this

and this is going to stand for uppercase a is a subset of uppercase B so in this

a is a subset of uppercase B so in this case both of them are

sets so in terms of Vin diagrams let's say the first one you should think of it

say the first one you should think of it like this so you have a blob like this

like this so you have a blob like this that's a and in it there's a

that's a and in it there's a point which is lowercase

point which is lowercase a the other one is more like this this

a the other one is more like this this is upper case

b and then here is uppercase a

okay again I don't want to uh talk too much about these things we're going to

much about these things we're going to keep using these symbols and class so

intervals so these are specific subsets of the real

line these are in a sense nice subsets of the real line

which recall we denoted by this R in Blackboard

Blackboard bold okay

bold okay so

so um I can be more abstract here but let

um I can be more abstract here but let me just give specific examples so for

me just give specific examples so for instance 1 comma 2 and I'm going to use

instance 1 comma 2 and I'm going to use these square brackets this means the set

these square brackets this means the set of all those

of all those numbers in rational or uh irrational

numbers in rational or uh irrational that is just say any real number that is

that is just say any real number that is between one and two end points included

between one and two end points included so I can write it like this using this

so I can write it like this using this set builder notation that we discussed

set builder notation that we discussed yesterday right so I need to open uh

yesterday right so I need to open uh braces close braces put a

braces close braces put a bar to the left I'm going to have to say

bar to the left I'm going to have to say something about the general form of the

something about the general form of the elements in the set so let's use some

elements in the set so let's use some letter um

X um sometimes you can also write it like this too X and R such

that this and then

this and then this okay nothing too complicated there

this okay nothing too complicated there is also the uh open interval so this is

is also the uh open interval so this is what is called the closed interval

the open interval is the same thing except that you don't include the end

except that you don't include the end points and here I'm going to use a uh

points and here I'm going to use a uh notation that is different from the

notation that is different from the notation that is used in the

notation that is used in the book so in the book they use this

book so in the book they use this notation let's use different numbers um

notation let's use different numbers um 3 to

3 to Pi Pi that you know the number 3.14 blah

Pi Pi that you know the number 3.14 blah blah blah um I'm going to use again

blah blah um I'm going to use again closed braces closed brackets but I'm

closed braces closed brackets but I'm going to turn them turn them the other

going to turn them turn them the other way

around okay so this is what is the what is called the French not

is called the French not notation uh people educated in France I

notation uh people educated in France I wasn't educated in France but still I I

wasn't educated in France but still I I like using this uh they like using this

like using this uh they like using this notation and the more American notation

notation and the more American notation is this of course okay I'm not going to

is this of course okay I'm not going to use this because this looks like a pair

use this because this looks like a pair this this looks like a point in the

this this looks like a point in the plane this is more clear to me and so

plane this is more clear to me and so what is

what is this this is a set of all those reals

this this is a set of all those reals such

such that

that well

well um X is strictly between 3 and pi

interval and the sort of intuition behind this notation is the following so

behind this notation is the following so you're sort of imagining this something

you're sort of imagining this something like a Lego piece okay

like a Lego piece okay so let's say you have well let me just

so let's say you have well let me just draw it um so here is three here is

pi okay then this is a notation the point

okay then this is a notation the point being is that to the left of it what you

being is that to the left of it what you have is let's say minus infinity comma

have is let's say minus infinity comma 3 what you see it is so the braces are

3 what you see it is so the braces are toward three here for this

toward three here for this part and this piece sort of you can sort

part and this piece sort of you can sort of you know join with this piece if you

of you know join with this piece if you want it that's sort of the

want it that's sort of the intuition any questions about

this is this too is this too straightforward to too

straightforward to too easy don't worry we're going to get

easy don't worry we're going to get things are going to get more complicated

things are going to get more complicated but still it's uh the second day of

but still it's uh the second day of classes okay so these are the two types

classes okay so these are the two types of intervals there are more of course

of intervals there are more of course um you can do half open half

um you can do half open half closed

closed um let's just write one of these

signify so we have a Clos inter on interal on yeah right so all numbers

interal on yeah right so all numbers between seven and n nine

between seven and n nine included exactly

included exactly so

so this sometimes people also use this

this sometimes people also use this notation by the way to emphasize that

notation by the way to emphasize that you're not including this so it's a

you're not including this so it's a strict

strict inequality and like

inequality and like so okay and there's a bunch of other

so okay and there's a bunch of other things uh but these are sort of the the

things uh but these are sort of the the standard

standard things any questions about these so far

okay set operations I'm just going to use intervals because it's more concrete

use intervals because it's more concrete um

operations let me just list them down and then we can write down a couple

and then we can write down a couple examples so what we have are the

examples so what we have are the following um we

following um we have this this

then this and

then this and then I wrote subset but still it's

still it's good so this stands for

intersection and the way you should think about it is that there's an N here

think about it is that there's an N here see it's coming down this stands for a

see it's coming down this stands for a union in u

this is what is called symmetric uh difference or symmetric

complement and this is um subset of of course

in a second I'm going to give examples of these now these um are somewhat they

of these now these um are somewhat they correspond each of these corresponds to

correspond each of these corresponds to a certain logical operation that we

a certain logical operation that we talked about before so maybe I can ask

talked about before so maybe I can ask you can you come up with a logical

you can you come up with a logical operation that corresponds

operation that corresponds intersection and yeah

right how about Union or exactly

Union or exactly this is hard to see but this is going to

this is hard to see but this is going to be xor exclusive

or so this is something like the following um so you want to say oh I'm

following um so you want to say oh I'm taking uh linear algebra or so let's say

taking uh linear algebra or so let's say the

the following let's say that this set

following let's say that this set corresponds this this set represents the

corresponds this this set represents the students who are taking linear

students who are taking linear algebra

algebra um and then this set represents the

um and then this set represents the students who are taking differential

students who are taking differential equations let's say it's a different

equations let's say it's a different class

class um and so the students who are taking

um and so the students who are taking linear algebra or differential equations

linear algebra or differential equations is the whole thing okay whereas the

is the whole thing okay whereas the students who are taking linear algebra

students who are taking linear algebra xor differential equations is not the

xor differential equations is not the whole thing

whole thing it's the students who are taking linear

it's the students who are taking linear algebra but not differential

algebra but not differential equations and differential equations but

equations and differential equations but not linear algebra okay so you sort of

not linear algebra okay so you sort of get rid of the intersection

get rid of the intersection there how about complements what is The

there how about complements what is The Logical operation that corresponds to

how about this subset of there's a logical operation it turns out that

logical operation it turns out that corresponds to the operation of or

corresponds to the operation of or relation of

relation of um being a

subset so something like this imagine let's say

imagine let's say that um so see that the smaller blob is

that um so see that the smaller blob is a subset of the larger

a subset of the larger blob that means that if someone is in

blob that means that if someone is in the smaller blob it's also in the larger

the smaller blob it's also in the larger blob so this corresponds to

okay I know the this is somewhat more abstract let's get more conc concrete

abstract let's get more conc concrete and give examples of

and give examples of these so for instance

um I'm just going to use intervals for all of this let's start with one

all of this let's start with one two in fact let me delete this as

well okay now what I can do is the following instead of writing two

following instead of writing two inequalities like this I can use the uh

inequalities like this I can use the uh connector and so I can say this is equal

connector and so I can say this is equal to the set of all those X's such

to the set of all those X's such that X is not less than one and X is not

that X is not less than one and X is not greater than two so

okay I mean writing this literally means the same thing as this but when you have

the same thing as this but when you have this this is the same thing as saying

this this is the same thing as saying this is the intersection this set is is

this is the intersection this set is is the intersection

the intersection of X such

of X such that X is not less than

that X is not less than one enter intersection X such that X is

one enter intersection X such that X is not uh greater than two

so this is what I mean when I say intersection in terms of set operations

intersection in terms of set operations correspond to and in terms of logical

correspond to and in terms of logical operations inside the bra bases I had

operations inside the bra bases I had the logical operation and then I was

the logical operation and then I was able to in a sense distribute right and

able to in a sense distribute right and turn it turn this logical operation into

turn it turn this logical operation into an

an intersection is this

intersection is this clear this is a silly thing in a sense

clear this is a silly thing in a sense I'm just like rewriting the same thing

I'm just like rewriting the same thing but it's going to be very very

but it's going to be very very helpful and then I can go further and

helpful and then I can go further and use the following notation for these two

use the following notation for these two and even though these are sort of

and even though these are sort of open-ended in a sense I still want to

open-ended in a sense I still want to think of them as intervals so I'm going

think of them as intervals so I'm going to write them like

this so this is the first piece one to infinity infinity you see not included

infinity infinity you see not included later on I think we're going to discuss

later on I think we're going to discuss a little bit what is called the extended

a little bit what is called the extended reals where you get to pretend that this

reals where you get to pretend that this Infinity is a number as

Infinity is a number as well um okay

well um okay intersection and then minus infinity to

two Okay so this set is the intersection of this and this

okay so uh let's look at an example with a

a union well um let's still stick to one

union well um let's still stick to one over one 1A

over one 1A 2 so I want to write one comma two as a

2 so I want to write one comma two as a union of two things let's

union of two things let's say um

so that's the first example the next example is the

2 well I can chop it in half somewhere let's say 1.5 and I can write it like

let's say 1.5 and I can write it like this

okay so visually what is happening here is the

from 1 to two that means that you have every single number in

every single number in between and I you know randomly chose

between and I you know randomly chose 1/2 oh sorry

1/2 oh sorry 1.5 so it's

1.5 so it's here now what does this notation mean it

here now what does this notation mean it means that I'm looking at all the

means that I'm looking at all the numbers between 1 and 1 12 one is

numbers between 1 and 1 12 one is included 1/2 is not included if it's not

included 1/2 is not included if it's not included here I must include it there

included here I must include it there and I hope this sort of graphical sort

and I hope this sort of graphical sort of thing also makes sense this is sort

of thing also makes sense this is sort of like a Lego piece you see you can

of like a Lego piece you see you can sort of like join these two things and

sort of like join these two things and you get this thing okay and so

you get this thing okay and so um this is 1.5 and you don't in you

um this is 1.5 and you don't in you don't you're not including it on the

don't you're not including it on the left side which means that you have to

left side which means that you have to include it on the right

include it on the right side does this make sense of course this

side does this make sense of course this is not unique right like I I can keep

is not unique right like I I can keep going like this and uh when we start

going like this and uh when we start talking about integrals for instance um

talking about integrals for instance um this partitioning an interval is going

this partitioning an interval is going to be an important thing one thing I can

to be an important thing one thing I can do for instance here I can do the

sorry one and then this and

then let's say that I didn't include it there either well I can just add it as a

there either well I can just add it as a as an uh as an additional set

it's the same thing the order in which you write these things doesn't matter

you write these things doesn't matter okay and that's one of the important

okay and that's one of the important things about set

operations does this make sense okay symmetric Union

um so for instance let's look at the symmetric un sorry symmetric difference

symmetric un sorry symmetric difference should

be it turns out that you can really think of this as

think of this as subtraction um let's look at 13 Clos

subtraction um let's look at 13 Clos interval symmetric

interval symmetric difference 2

four now when I have this I want to have all those

all those numbers that are in either this but not

numbers that are in either this but not this or in this but not that so what is

this or in this but not that so what is the intersection here let's draw

it so one 2 3 4 so the uh first interval is the

4 so the uh first interval is the this the second interval is

this the second interval is this

okay I'm looking at exclusive or well that means that I need to get

or well that means that I need to get rid of this middle piece the

rid of this middle piece the intersecting

intersecting piece so it's going to be 1 two and then

piece so it's going to be 1 two and then 3 4

okay yes since three and two are inclusive in that does that mean they

inclusive in that does that mean they shouldn't

shouldn't be um you're right thank

be um you're right thank you what your colleague is correctly

you what your colleague is correctly correcting me is

correcting me is that the intersection

that the intersection here has

here has the point two in it right because two is

the point two in it right because two is in this part and also it's in this part

in this part and also it's in this part which means that it cannot be in the

which means that it cannot be in the xord

part okay does this make sense okay uh complement

well um maybe let's write it like this one to Infinity let's say one

one to Infinity let's say one included to the C that's the notation

included to the C that's the notation there's another notation which is this

there's another notation which is this you write the name of the larger sets

you write the name of the larger sets and then you put a uh

and then you put a uh Slash and then you write the set

Slash and then you write the set so this is the set of all those points

so this is the set of all those points in the reals that are not in this set

in the reals that are not in this set which means this should be the

which means this should be the following what is

remaining see the this is the missing Lego piece if you have the whole real

Lego piece if you have the whole real line and if you were to take out this

line and if you were to take out this Lego piece you're left with this Lego

Lego piece you're left with this Lego piece

okay and then finally maybe let's write something for uh

subset um one two well

closed okay now how do you interpret this from

okay now how do you interpret this from a logical point of view if you want to

a logical point of view if you want to think of this logically what this is is

think of this logically what this is is saying is the following silly statement

saying is the following silly statement really if I have a number that is

really if I have a number that is strictly between 1 and

strictly between 1 and two it could also be equal to one or two

two it could also be equal to one or two obviously that's not going to happen but

obviously that's not going to happen but if it's here that's satisfied also that

if it's here that's satisfied also that is to say you can always relax strict

is to say you can always relax strict inequalities to non-strict inequalities

um well that is not always true in this case it is true but it's not always true

case it is true but it's not always true so I'm not going to write

so I'm not going to write it

it okay so that's all there is to know

okay so that's all there is to know about sets

about sets I want to talk about something called um

I want to talk about something called um cartisian product and

cartisian product and relation and then we're going to be able

relation and then we're going to be able to start talking about functions I don't

to start talking about functions I don't want to push it though so I think it's a

want to push it though so I think it's a good place to stop today thank you I'll

good place to stop today thank you I'll see you

tomorrow okay good morning um so today we're going to keep uh working on um set

we're going to keep uh working on um set theory stuff certain constructions that

theory stuff certain constructions that we're going to use throughout this class

we're going to use throughout this class um but before that I wanted to mention

um but before that I wanted to mention this at the end of last class one of

this at the end of last class one of your colleagues came to me and mentioned

your colleagues came to me and mentioned the following that in the textbook that

the following that in the textbook that they were looking into as well uh the

they were looking into as well uh the notation that they use for um a being a

notation that they use for um a being a subset of B is this instead of this okay

subset of B is this instead of this okay and

and so recall this was the uh notation that

so recall this was the uh notation that I introduced

yesterday so this stood for A and B are subsets A and B are sets and a is a

subsets A and B are sets and a is a subset of B

so and the picture was this right so we have this larger blob that I uh call B

have this larger blob that I uh call B and in it there's a smaller blob

and in it there's a smaller blob maybe it has multiple parts and stuff

maybe it has multiple parts and stuff like that but for visualization purposes

like that but for visualization purposes it's nice to do

it's nice to do this now the reason why I use this

this now the reason why I use this notation as opposed to this the idea is

notation as opposed to this the idea is that I'm sort of thinking of this in

that I'm sort of thinking of this in analogy with this less than or equal to

analogy with this less than or equal to sign when it comes to

numbers okay so this is kind of like this um now in the book they mean

this um now in the book they mean exactly the same same thing as this when

exactly the same same thing as this when they use this notation but I'm never

they use this notation but I'm never going to use this notation okay I'm

going to use this notation okay I'm always going to uh put this and the

always going to uh put this and the difference here is the

following just like here you have strict inequality remember from yesterday that

inequality remember from yesterday that I said sometimes people write this and

I said sometimes people write this and they put sort of like a dash so they

they put sort of like a dash so they cross the sort of the equal part of this

cross the sort of the equal part of this uh symbol to signify that it's a strict

uh symbol to signify that it's a strict inequality and likewise here this is

inequality and likewise here this is going to represent a strict subset so in

going to represent a strict subset so in this case a might well be the same thing

this case a might well be the same thing as B so let me write it um in this

notation a might be b

equal to B and of course uh let's also mention what

and of course uh let's also mention what it means for two sets to be equal well

it means for two sets to be equal well two sets are equal if uh any element in

two sets are equal if uh any element in one of them is an element in the other

one of them is an element in the other one and vice versa right so sets are

one and vice versa right so sets are supposed to be mathematical objects that

supposed to be mathematical objects that are amorphous they they don't have any

are amorphous they they don't have any structure whatsoever later on we're

structure whatsoever later on we're going to learn certain structures uh

going to learn certain structures uh topological or continuous structure and

topological or continuous structure and then a smooth

then a smooth structure but when it comes to sets they

structure but when it comes to sets they are defined precisely by what is in them

are defined precisely by what is in them and what is not in

and what is not in them okay so maybe I should give an

them okay so maybe I should give an example here as well um so I can write

example here as well um so I can write something like

something like this so for

this so for instance

instance R is a

R is a subset of

itself it's more of course R is equal to itself but certain that this is also

itself but certain that this is also true and let me perhaps write the write

true and let me perhaps write the write this as well

um this is false okay because this is saying this thing is a what is called

saying this thing is a what is called proper subset it's a subset but it is

proper subset it's a subset but it is not equal to okay so this is a false

statement any questions about this now there's a further perhaps more

now there's a further perhaps more philosophical question now if it is the

philosophical question now if it is the case that in the textbook they're using

case that in the textbook they're using this notation why am I using this

this notation why am I using this notation right well I kind of mentioned

notation right well I kind of mentioned why uh this is sort of the idea because

why uh this is sort of the idea because I'm using this symbol in analogy with

I'm using this symbol in analogy with this less than or equal to

this less than or equal to sign um another thing too of course is

sign um another thing too of course is the fact that this or this or some other

the fact that this or this or some other notation these are not Universal

notation these are not Universal notations okay okay so from textbook to

notations okay okay so from textbook to Tex textbook it changes and um often

Tex textbook it changes and um often from field to field it changes also and

from field to field it changes also and so this is also part of the uh part of

so this is also part of the uh part of developing this mathematical maturity so

developing this mathematical maturity so um so one of the issues with sort of

um so one of the issues with sort of sticking to one textbook is that it

sticking to one textbook is that it gives a false sense of security when it

gives a false sense of security when it comes to what people call things how

comes to what people call things how people denote things and so on and then

people denote things and so on and then then you uh overfit your understanding

then you uh overfit your understanding right you so you you're sort of like

right you so you you're sort of like trying to understand something and then

trying to understand something and then you over specify and then you go out

you over specify and then you go out into the real world and let's say you're

into the real world and let's say you're trying to do some engineering problem or

trying to do some engineering problem or some other mathematical thing let's say

some other mathematical thing let's say and then you come across a paper another

and then you come across a paper another book and so on and then you realize you

book and so on and then you realize you cannot read it because none of the

cannot read it because none of the symbols are similar or familiar to you

symbols are similar or familiar to you and I'm trying to combat that uh from

and I'm trying to combat that uh from the get-go and you know it is the case

the get-go and you know it is the case that in mathematics or in uh so whenever

that in mathematics or in uh so whenever you're doing you're looking into

you're doing you're looking into something that uses mathematics

something that uses mathematics engineering physics and all that stuff

engineering physics and all that stuff um it turns out that um different people

um it turns out that um different people call same things different names and

call same things different names and sometimes different things the same

sometimes different things the same thing okay so it it depends on the

thing okay so it it depends on the context and stuff like that so I'm just

context and stuff like that so I'm just I I just want to prepare you about

I I just want to prepare you about this of course it makes things a little

this of course it makes things a little bit more complicated but I think it is

bit more complicated but I think it is worthwhile

this now then there's the question of oh if that is the case should I even be uh

if that is the case should I even be uh looking into the textbook I would say

looking into the textbook I would say yes I would

yes I would say try reading the textbook of course

say try reading the textbook of course you're going to have to come to the

you're going to have to come to the classes also because grading the um

classes also because grading the um attendances and try to reconcile all the

attendances and try to reconcile all the time the things that are done in the

time the things that are done in the textbook and the way I'm doing things so

textbook and the way I'm doing things so here this was an instance of notation

here this was an instance of notation being different later on starting from

being different later on starting from next week in fact we're going to have

next week in fact we're going to have even more uh dramatic uh discrepancies

even more uh dramatic uh discrepancies one of them will be the following so for

one of them will be the following so for instance in the textbook uh um they

instance in the textbook uh um they spend a lot of time building the axioms

spend a lot of time building the axioms and proving things uh from bottom up and

and proving things uh from bottom up and stuff like that I'm not going to spend

stuff like that I'm not going to spend that much time I'm just going to say a

that much time I'm just going to say a lot of things I'm just going to take a

lot of things I'm just going to take a lot of things as axioms because I draw

lot of things as axioms because I draw to spend more time proving things about

to spend more time proving things about integrals and uh derivatives and other

integrals and uh derivatives and other things and so you're going to see things

things and so you're going to see things that I just say as axum I just State as

that I just say as axum I just State as axioms in class but in the textbook

axioms in class but in the textbook they're just like trying to prove it and

they're just like trying to prove it and stuff like that so that's also part of

stuff like that so that's also part of this exercise about uh reconciling two

this exercise about uh reconciling two accounts and this is also the thing that

accounts and this is also the thing that that's sort of like a standard thing

that's sort of like a standard thing when it comes to mathematics you say you

when it comes to mathematics you say you see in one book an exercise and it turns

see in one book an exercise and it turns out that the exercise in one book is

out that the exercise in one book is actually the definition in the other

actually the definition in the other book and so on so there's no unified

book and so on so there's no unified account of things and you should get

account of things and you should get used to these

things that's all I wanted to say about this any questions about

this any questions about this

this okay okay um so I'd like to mention

okay okay um so I'd like to mention three ways of building new sets out of

three ways of building new sets out of old ones

old ones and um there are certain levels at which

and um there are certain levels at which you can discuss these things I'm going

you can discuss these things I'm going to not go too much into detail I'm not

to not go too much into detail I'm not going to try to be too formal about

going to try to be too formal about these these things uh I'm just going to

these these things uh I'm just going to Define them and uh we're going to be on

Define them and uh we're going to be on our way so um the first method is what

our way so um the first method is what is called cartisian

is called cartisian Product okay so this is a way of

Product okay so this is a way of multiplying

multiplying sets uh cartisian

sets uh cartisian because um

because um allegedly day cart one of the sort of

allegedly day cart one of the sort of modern philosophers or probably the

modern philosophers or probably the first modern philosophers if you ask ask

first modern philosophers if you ask ask someone some people um he was the one

someone some people um he was the one who realized that oh you if you have an

who realized that oh you if you have an input output relation well what you can

input output relation well what you can do is you can um turn it into a um

do is you can um turn it into a um coordinate system and sort of draw

coordinate system and sort of draw things in a

things in a plane and so that's why it is called the

plane and so that's why it is called the cartisian product

the second is what is called Power set and this is sort of like a method of

and this is sort of like a method of going to The Meta I'm going to discuss

going to The Meta I'm going to discuss what that means in a second and then the

what that means in a second and then the third one is um looking at a set of

third one is um looking at a set of functions or sets of functions between

functions or sets of functions between sets okay it's going to be another set

sets okay it's going to be another set of course

of course but um its elements are going to be it's

but um its elements are going to be it's going to be important to think of the

going to be important to think of the elements of the set as not only points

elements of the set as not only points but also Al things that you can

but also Al things that you can graph and this is where things are uh

graph and this is where things are uh starting to get a little bit more

starting to get a little bit more abstract of course so method

abstract of course so method one so maybe I should say just

one so maybe I should say just definition

definition um let a um X and Y

sets then it's cartisian then their cartisian product is defined as

set I'm going to denote it by x * y or X cross

Y and this will be the set of all those pairs such that the first element in the

pairs such that the first element in the pair comes from X and the second element

pair comes from X and the second element comes from y

comes from y and so I can use this set builder

and so I can use this set builder notation

notation okay open curly

okay open curly braces and then bar and then Clos curly

braces and then bar and then Clos curly braces here I need to write down the

braces here I need to write down the typical form so it's going to be

typical form so it's going to be lowercase x comma lowercase Y in

lowercase x comma lowercase Y in parenthesis so whenever I write it like

parenthesis so whenever I write it like this this is an ordered Tuple

this this is an ordered Tuple okay such that on the right hand side I

okay such that on the right hand side I need to write down the condition x

need to write down the condition x lowercase x comes from uppercase X and

lowercase x comes from uppercase X and lowercase y comes from uppercase

okay what's the first example of course uh ukian spaces or powers of r

method let me just write it here um so for

um so for instance what you would denote by R to

instance what you would denote by R to the 2 right the um plane

the 2 right the um plane well this is nothing but R *

R so in a sense this is R squ that's sort of the the

idea any questions about this yes so is it genuinely a cross product or just

it genuinely a cross product or just multiplication it's

multiplication it's um I

um I mean

mean um you mean in terms of vectors yeah no

um you mean in terms of vectors yeah no it's not a vector it's not it's not a

it's not a vector it's not it's not a it's not a cross product that you would

it's not a cross product that you would learn in multivariable calculus but it

learn in multivariable calculus but it is denoted by a

is denoted by a cross third thing that we using the

cross third thing that we using the little cross yeah yeah I mean there are

little cross yeah yeah I mean there are only so many

only so many symbols

symbols right and uh I mean yeah there's this

right and uh I mean yeah there's this thing there's this um constant struggle

thing there's this um constant struggle because there are only so many things so

because there are only so many things so many symbols and so when you try to

many symbols and so when you try to represent a certain operation either

represent a certain operation either either you're going to use something

either you're going to use something that was used to sort of like signify oh

that was used to sort of like signify oh it is like this it's like I'm just

it is like this it's like I'm just multiplying R by itself and I get the r

multiplying R by itself and I get the r squ or sometimes you want to say oh but

squ or sometimes you want to say oh but it's not exactly like this and that's

it's not exactly like this and that's why for instance there are similar

why for instance there are similar symbols where you put things like

symbols where you put things like um cross but in a circle and stuff like

um cross but in a circle and stuff like that this is what is called the tensor

that this is what is called the tensor product it's like a it's just it's just

product it's like a it's just it's just a product

but this is fairly Universal this notation

notation okay any other questions about

this okay um okay so that's the first method

okay um okay so that's the first method second method is what is called the

second method is what is called the power

set let X be a set then its power set which is denoted by P

then its power set which is denoted by P of X is the set of all of x's subsets

of X is the set of all of x's subsets and so things are getting complicated

here then it's power

set is defined to be

following set so I'm going to denote it by P of X so I like to use this curly uh

by P of X so I like to use this curly uh p p of x equals so how do I write this

p p of x equals so how do I write this again I can use a set builder

notation typical element is a and the condition is a is a sub subset of x

okay so this means that using this power set um operation you can relate this

set um operation you can relate this element of relation that we discussed

element of relation that we discussed yesterday and the subset of

yesterday and the subset of relation and so

X here's a new symbol this is how I'm going to how I'm going to read this is

going to how I'm going to read this is I'm going to say if and on if it is

I'm going to say if and on if it is equivalent so it implies this implies

equivalent so it implies this implies this side and this implies this side um

this side and this implies this side um this if and only if a is an

this if and only if a is an element of X of P of

X Let's do an example and in a second I would like to say uh make a comment

I would like to say uh make a comment about why it's called the power set as

about why it's called the power set as well so here's an example

well so here's an example um when you're trying to understand

um when you're trying to understand these kind of things it's nice to think

these kind of things it's nice to think about a finite set okay so for instance

about a finite set okay so for instance let's look at

let's look at this let me let X to be the

this let me let X to be the set consisting of two things two letters

set consisting of two things two letters A and B what is its power set its power

A and B what is its power set its power set is supposed to be the set of all

set is supposed to be the set of all subsets in X okay so I need to list all

subsets in X okay so I need to list all of

of them

them so then P of X is well I need to start

so then P of X is well I need to start with what is called empty sets this is a

with what is called empty sets this is a silly math thing but you have to include

silly math thing but you have to include the thing that is the most you know

the thing that is the most you know basic most obvious thing this so o with

basic most obvious thing this so o with a with a dash this is the empty set

a with a dash this is the empty set empty set has nothing in it and then you

empty set has nothing in it and then you have a set that consists only of a in

have a set that consists only of a in little set that consists only of B and

little set that consists only of B and then exits

itself okay so maybe I should denote it here oh okay yeah sorry uh I'll be more

here oh okay yeah sorry uh I'll be more careful going forward

here this thing is it STS it stands for empty set or the null set and

stuff and how do you denote this well you just open braces and close braces

you just open braces and close braces without without writing anything

yes we not including a set of all

um yeah maybe

maybe sorry but you're right you include all

sorry but you're right you include all of them and you see the the interesting

of them and you see the the interesting thing here is that this set X is an

thing here is that this set X is an element in P of X now why is it called

element in P of X now why is it called why is p ofx called the power set well

why is p ofx called the power set well you look at um

you look at um how the number of

how the number of things in the set change okay so for

things in the set change okay so for instance in this case X has two elements

instance in this case X has two elements and P of X has four elements okay so if

and P of X has four elements okay so if I had three elements in X let's say x

I had three elements in X let's say x consists of a b and c how many elements

consists of a b and c how many elements do you think would have would uh P of x

do you think would have would uh P of x uh how many Elms would do you think

uh how many Elms would do you think would it have yeah

yes write exactly exactly that's a good

write exactly exactly that's a good point

point so maybe I should uh rewrite it

here this is the same thing I was writing oh

writing this okay now why is that well it's because X this capital x is

it's because X this capital x is literally the same thing as this and

literally the same thing as this and then it comes to

then it comes to sets like they they are the same thing

sets like they they are the same thing so you can just substitute whenever you

so you can just substitute whenever you see one instead of the one I was just

see one instead of the one I was just being

being lazy

lazy okay okay so back to the question let's

okay okay so back to the question let's say that x capital x has three elements

say that x capital x has three elements how many elements does p of X

have yes right so I'm going to leave that as an

so I'm going to leave that as an exercise that's a good point if it is

exercise that's a good point if it is not clear to you what you should do is

not clear to you what you should do is you should write it out explicitly it's

you should write it out explicitly it's going to be painful but it is useful

going to be painful but it is useful okay write it out explicitly list all of

okay write it out explicitly list all of the

let me write it like this number of things in

things in X I'm just going to denote it by by

implies what is 8 of course let's do some numerology how do you relate 8 to

some numerology how do you relate 8 to three it is 2 to the 3 let's write it

three it is 2 to the 3 let's write it like

that and so this is the this is my notation for number of things in X

notation for number of things in X number of

elements okay you can write it out explicitly say

okay you can write it out explicitly say x contains a b and c write all of the

elements the main idea is the following and this calculation and this is going

and this calculation and this is going to lead to the next exercise and then

to lead to the next exercise and then I'm giving these exercises in so far as

I'm giving these exercises in so far as they're not in the problem sets you

they're not in the problem sets you don't need to turn it in okay so these

don't need to turn it in okay so these is these are just things for you to

is these are just things for you to think about but of course I might sort

think about but of course I might sort of think about them and put them in the

of think about them and put them in the problem set so it is in your benefit

problem set so it is in your benefit it's beneficial for you to think think

it's beneficial for you to think think about them just in case and they might

about them just in case and they might show up in the exam as

show up in the exam as well

well um the idea is this let's say you start

um the idea is this let's say you start with your capital x that's your initial

with your capital x that's your initial set now you're trying to understand the

set now you're trying to understand the power set of it of it so an element in P

power set of it of it so an element in P of X is a subset in X how do you build a

of X is a subset in X how do you build a subset in X well you look at any element

subset in X well you look at any element in X and you decide is this element

in X and you decide is this element going to be in my subset or not okay so

going to be in my subset or not okay so that means that for any element in X you

that means that for any element in X you have two choices either you include it

have two choices either you include it or you don't in the subset and based on

or you don't in the subset and based on that you have you know 2 * 2 * 2

that you have you know 2 * 2 * 2 ultimately and in fact it turns out that

ultimately and in fact it turns out that this is more generally

true this is equal to the to two to the number of

number of things in

things in X and this is why it's called the power

X and this is why it's called the power set I mean presumably I don't know why

set I mean presumably I don't know why it is called Power set but this seems to

it is called Power set but this seems to be like one way you could

justify okay now it turns out that this equation is true even when X is not a

equation is true even when X is not a finite

finite set so maybe you want to take X to be

set so maybe you want to take X to be the set of integers which is an infinite

the set of integers which is an infinite set and then okay so it's kind of

set and then okay so it's kind of dubious what it means uh so it's kind of

dubious what it means uh so it's kind of dubious to use the word number of things

dubious to use the word number of things when it comes to an infinite set that's

when it comes to an infinite set that's why people use this um phrase called

why people use this um phrase called cardinality so if you want to be more

cardinality so if you want to be more proper you say

cardinality and it turns out that for instance um

instance um in a sense the number of subsets of the

in a sense the number of subsets of the integers is two to the infinity many

integers is two to the infinity many very Infinity is the number of

very Infinity is the number of integers and there are different types

integers and there are different types of Infinities and stuff like that um

of Infinities and stuff like that um we're not going to spend that much time

we're not going to spend that much time on this stuff but it turns out that it

on this stuff but it turns out that it is somewhat useful maybe if it turns out

is somewhat useful maybe if it turns out that uh we're going to use them uh we

that uh we're going to use them uh we might sort of discuss it a little bit

might sort of discuss it a little bit more but it turns out let me just leave

more but it turns out let me just leave it at that for now there are different

it at that for now there are different types of Infinities and all that

okay good so next up there's a third method but for that to to discuss the

method but for that to to discuss the third method I need to tell you what the

third method I need to tell you what the function is and even before that I want

function is and even before that I want to do something slightly different and I

to do something slightly different and I want to discuss um this other notion

want to discuss um this other notion which is more General than a function U

which is more General than a function U it's called a relation how many of you

it's called a relation how many of you have heard this term

have heard this term relation okay couple of you

so oh maybe I should ask as well how many of you have heard the term

many of you have heard the term function okay you took calculus after

function okay you took calculus after all okay

now the notion of a relation is a notion that is more relaxed relative to um the

that is more relaxed relative to um the notion of a function and it's all over

notion of a function and it's all over the place okay and it's it's good to

the place okay and it's it's good to uh it's good to see it at Le

once and we're going to use it later as well so you might as well talk about it

well so you might as well talk about it now now now that we're talking about the

now now now that we're talking about the foundations and all

sets and of course I might as well to I I can take them to be the same set but I

I can take them to be the same set but I don't need to

then now the definition is going to be very CH

very CH okay then a relation from X to

Y is uh is a subset of their cartisian

product is by

is by definition in fact let me do this I'm

definition in fact let me do this I'm not going to keep writing by definition

not going to keep writing by definition whenever I write whenever I

whenever I write whenever I underline um a phrase in a sentence that

underline um a phrase in a sentence that means that I'm defining that that phrase

means that I'm defining that that phrase is a

relation from X to Y if and only if R is an

an element of the power set of x * y

um let's start like this I want to be concrete

this I want to be concrete here let's take X to be the unit

here let's take X to be the unit interval you know Clos interval from 0

interval you know Clos interval from 0 to

to one and I'm going to take y to be the

one and I'm going to take y to be the same thing

the benefit of doing this of course is that I can also draw it right

so I'm going to give a couple examples here let me take X and Y to be both

here let me take X and Y to be both 01 I can certainly do this now what is x

01 I can certainly do this now what is x *

y geometrically speaking so if both of these sets is a line segment what is x *

these sets is a line segment what is x * y yeah it's a

y yeah it's a square

square um so how would you write this this

um so how would you write this this is if I were to substitute 01 instead of

is if I were to substitute 01 instead of X and

X and Y this is how you would write and of

Y this is how you would write and of course if you wanted to draw these what

course if you wanted to draw these what you would do is the

you would do is the following here is both x and y and x * y

following here is both x and y and x * y is just this

guy okay now when I'm talking about a relation um on X I'm not going to say

relation um on X I'm not going to say why because they're the same

why because they're the same thing I said that a relation is a subset

thing I said that a relation is a subset of the cross product of the of the

of the cross product of the of the cartisian product that is to

cartisian product that is to say a relation R is is going to be a

say a relation R is is going to be a subset of this Square well geometrically

subset of this Square well geometrically speaking a subset is just a

speaking a subset is just a shape so a

X is a shape in the

okay so for instance here's one

relation okay so this is a subset of x * X which means that which means that a

X which means that which means that a typical element in R is a pair of

typical element in R is a pair of numbers and also note that even though

numbers and also note that even though we're looking into um single variable

we're looking into um single variable calculus in this class immediately we

calculus in this class immediately we have two variables and in fact um we're

have two variables and in fact um we're going to have infinite Dimensions uh

going to have infinite Dimensions uh when we start looking into this function

when we start looking into this function space okay so let me denote let me uh

space okay so let me denote let me uh Define r to be the following X comma

that X is less than or equal to

is less than or equal to Y is this a subset certainly it is it is

Y is this a subset certainly it is it is well defined and I can even draw it and

well defined and I can even draw it and uh in fact probably some of you are

uh in fact probably some of you are familiar with this this type of thing

familiar with this this type of thing from integration in multivariable

from integration in multivariable calculus calculus and stuff so how would

calculus calculus and stuff so how would I draw this well here's my

I draw this well here's my Square here's my main

Square here's my main diagonal

diagonal um so X is supposed to be the horizontal

um so X is supposed to be the horizontal variable in this represent a and Y is

variable in this represent a and Y is supposed to be the vertical why is it

supposed to be the vertical why is it that way and what is the formalism

that way and what is the formalism behind it I mean let's not get too much

behind it I mean let's not get too much into these kind of things it's clear

into these kind of things it's clear that I'm sort of representing this as X

that I'm sort of representing this as X and this as y if you want to be more

and this as y if you want to be more specific you would just draw sort of

specific you would just draw sort of like the axis and name the axes and so

like the axis and name the axes and so on um okay so

on um okay so certainly the diagonal where x equals y

certainly the diagonal where x equals y should be included in my relation so the

should be included in my relation so the main diagonal is

um so so should I be including the upper triangle or the lower triangle triangle

triangle or the lower triangle triangle upper why because so let's test it out

upper why because so let's test it out um so if I were to plug in x

um so if I were to plug in x equals 12 which means that I'm testing

equals 12 which means that I'm testing this thing well I need to include in my

this thing well I need to include in my relation all those y's that are equal or

relation all those y's that are equal or greater than2 which means that in this

greater than2 which means that in this vertical

vertical slice I should be including the top part

slice I should be including the top part okay which means that I should be

okay which means that I should be including

including um this triangle over

now in this case things are rather uh perhaps obvious but even when you do

perhaps obvious but even when you do more and more abstract stuff it's good

more and more abstract stuff it's good to be able to switch from this sort of

to be able to switch from this sort of syntactic notation to some picture some

syntactic notation to some picture some caricature and vice versa and stuff and

caricature and vice versa and stuff and in fact you see I can think of this

in fact you see I can think of this which is a relation

which is a relation itself um so this I want to think of as

itself um so this I want to think of as as a relation in and of itself I can

as a relation in and of itself I can think of it as a subset I can think of

think of it as a subset I can think of something that relates two things and I

something that relates two things and I can think of it geometrically so this is

can think of it geometrically so this is an important thing to be able to

an important thing to be able to do okay so this I would say is the less

do okay so this I would say is the less than or equal to

[Applause] relation any questions about

this okay so let's get a little bit more

okay so let's get a little bit more abstract um can I think

abstract um can I think of um the notion of something being an

of um the notion of something being an element of something else as a

element of something else as a relation within this

formalism H maybe even before that let's do something more concrete again based

do something more concrete again based on calculus and then we can look at that

on calculus and then we can look at that problem um so here's another thing

relation um X comma y such that x^2 + y^2 = 1

this a certainly a relation how can I think of this

geometrically it's a circle right what you would say what you would call a

you would say what you would call a circle it's a shape after all is a

circle it's a shape after all is a relation it's the unit

circle okay so in a sense you were using

okay so in a sense you were using relations all along and you know that

relations all along and you know that this is not a function uh well maybe I

this is not a function uh well maybe I should ask how many of you know that

should ask how many of you know that this is not a

this is not a function how would you sort of justify

function how would you sort of justify it's not a

it's not a function there's like a test and stuff

function there's like a test and stuff it's a vertical line yeah right so

it's a vertical line yeah right so there's this thing called the vertical

there's this thing called the vertical line test which says that if you have a

line test which says that if you have a shape a

shape a relation it's a function it defines a

relation it's a function it defines a function if whenever you CH whenever you

function if whenever you CH whenever you draw a vertical line you hit whatever

draw a vertical line you hit whatever you you drew at exactly one point and in

you you drew at exactly one point and in this case you see um here's a line that

this case you see um here's a line that hits the the shape at two points now if

hits the the shape at two points now if I were to use this other line of course

I were to use this other line of course it hits the shape at one point but it

it hits the shape at one point but it doesn't matter for the vertical line

doesn't matter for the vertical line test

test um you should be hitting at exactly 1

um you should be hitting at exactly 1 point for any vertical

point for any vertical line okay so not a function but it's a

line okay so not a function but it's a relation and so

relation and so on now if you uh if you're one of those

on now if you uh if you're one of those who are going to take the second the SQL

who are going to take the second the SQL to this class there you know it's the

to this class there you know it's the multivariable calculus version of this

multivariable calculus version of this class um you're going to learn a theorem

class um you're going to learn a theorem which is very important is it's called

which is very important is it's called the implicit function theorem which

the implicit function theorem which which tells you when you can in fact

which tells you when you can in fact solve y for X so for instance in this

solve y for X so for instance in this case what you can do is you can take oh

case what you can do is you can take oh y is 1 - x^2 and square root but then

y is 1 - x^2 and square root but then maybe Plus square root minus square root

maybe Plus square root minus square root so it's not a function exactly but maybe

so it's not a function exactly but maybe you can choose a plus at some point and

you can choose a plus at some point and that kind of works and stuff like that

that kind of works and stuff like that so you're going to get to the uh

so you're going to get to the uh formalism behind it for now this is just

formalism behind it for now this is just where I want to leave it at and there

where I want to leave it at and there are many other things of course that you

are many other things of course that you can

can do any questions about these

do any questions about these things so let's uh go back to this more

things so let's uh go back to this more abstract thing

I want to think of the element of of

relation so what do I need to do I need to do

so what do I need to do I need to do something like the

following so I want the following to make sense

make sense so here's the set builder notation

again you see this is an important thing to be able to do and a lot of people for

to be able to do and a lot of people for instance uh people who are doing physics

instance uh people who are doing physics are very good at these kind of things at

are very good at these kind of things at least good ones they take a mathematical

least good ones they take a mathematical formalism and they overload it without

formalism and they overload it without worrying too much about whether or not

worrying too much about whether or not it makes sense and then they figure out

it makes sense and then they figure out that it makes sense or they leave it at

that it makes sense or they leave it at someone else to make to make sense of it

someone else to make to make sense of it so I can just keep following this set

so I can just keep following this set builder notation and see if I can make

builder notation and see if I can make sense of it

sense of it okay so I want to say similar to this

okay so I want to say similar to this situation over here but I want to use

situation over here but I want to use the element of

relation okay so substituting everything just you know without thinking too much

just you know without thinking too much instead of X here I see a lower case a

instead of X here I see a lower case a so instead of X here I should be seeing

so instead of X here I should be seeing a lowercase a and instead of Y I should

a lowercase a and instead of Y I should be saying seeing an proper case

a so that means that I should be thinking of it like

thinking of it like this which means that the first set here

this which means that the first set here should be some set X second should be

should be some set X second should be its power

its power set so this

set so this is a relation from

x x * P of X certainly is valid because I can take the cartisian product of any

I can take the cartisian product of any two sets whatsoever so it's all

two sets whatsoever so it's all good now it turns out that if you keep

good now it turns out that if you keep pushing these things and if you keep

pushing these things and if you keep sort of uh building these self

sort of uh building these self referential things and if you keep sort

referential things and if you keep sort of using these operations in conjunction

of using these operations in conjunction too many times for instance you can look

too many times for instance you can look at P of P of P of X which is valid um

at P of P of P of X which is valid um eventually it turns out that you can

eventually it turns out that you can break math there's a bunch of bugs and

break math there's a bunch of bugs and stuff and that's why there's a bunch of

stuff and that's why there's a bunch of like words about you know like sets

like words about you know like sets instead of saying set of sets you say

instead of saying set of sets you say collection of sets and stuff like that

collection of sets and stuff like that let's not get into that back in let's

let's not get into that back in let's say the 20th century last century people

say the 20th century last century people were very worried worried about these

were very worried worried about these kind of things the foundations of math

kind of things the foundations of math and stuff like that uh now now days

and stuff like that uh now now days especially from a more pragmatic point

especially from a more pragmatic point of view we realize that okay so if you

of view we realize that okay so if you push it too much you're going to break

push it too much you're going to break it don't don't push it too much then

it don't don't push it too much then it's like don't there it is not

it's like don't there it is not Limitless just we just know that and um

Limitless just we just know that and um when it comes to using math or proving

when it comes to using math or proving the types of things that we're going to

the types of things that we're going to prove it's not going to be too much of a

prove it's not going to be too much of a problem but there are sort of like

problem but there are sort of like certain issues

certain issues underneath okay now similarly oh even

underneath okay now similarly oh even before this maybe I should leave it as

before this maybe I should leave it as an exercise to try to

an exercise to try to visualize this element of relation for

visualize this element of relation for our

our set

um let's say AB

and let's say capital A is well the set that contains only

that contains only a try to visualize this guy exercises

that oh I guess I need to uh not fix out Cas say

sorry now how can I start doing this well I need to draw I need to somehow

well I need to draw I need to somehow draw x * P of X well these are finite

draw x * P of X well these are finite sets in this case so I can just you know

sets in this case so I can just you know put a bunch of points so um this is the

put a bunch of points so um this is the first coordinate X has two

first coordinate X has two things and then P of X has four things

things and then P of X has four things which means it's going to be or it's

which means it's going to be or it's going to be something like this right so

going to be something like this right so um

so each of these Corners represent one points and then you can draw this

points and then you can draw this element of relation okay so of course

element of relation okay so of course you need to label these terms and stuff

you need to label these terms and stuff like

like that okay you see you take this idea of

that okay you see you take this idea of being able to think of something

being able to think of something geometrically and you apply it to

geometrically and you apply it to something more abstract still it's not

something more abstract still it's not very abstract but still it's it's it's

very abstract but still it's it's it's it's nontrivial to go from this to this

it's nontrivial to go from this to this okay but it is very

do any questions about yes you've drawn a cartisian plane um no no I mean if you

a cartisian plane um no no I mean if you want to be perhaps more accurate you

want to be perhaps more accurate you don't want to put the lines there but

don't want to put the lines there but putting the lines there is easier to

putting the lines there is easier to sort of draw because in this case X time

sort of draw because in this case X time P of X is just going to be a bunch of

points and we haven't talked about this but like I said before sets are supposed

but like I said before sets are supposed to be amorphous things so for instance

to be amorphous things so for instance when I'm writing when I'm drawing this

when I'm writing when I'm drawing this picture I'm in fact assuming certain

picture I'm in fact assuming certain structure that allows me to actually

structure that allows me to actually take this syntactic representation and

take this syntactic representation and draw it like this so it's a vector space

draw it like this so it's a vector space blah blah blah there's a bunch of things

blah blah blah there's a bunch of things that I'm saying I'm doing

that I'm saying I'm doing okay in this case in this more abstract

okay in this case in this more abstract exercise example there are no sort of

exercise example there are no sort of structures like that but still I can

structures like that but still I can pretend pretend that there is some

pretend pretend that there is some structure just to

structure just to visualize

questions now here's a definition

um and this is going to lead to the Third Way of defining a new set I'm

Third Way of defining a new set I'm going to say that a relation is the

going to say that a relation is the graph of a function if a certain thing

graph of a function if a certain thing happens well I'm just going to take this

happens well I'm just going to take this idea of this vertical line set and

idea of this vertical line set and Abstract it out

let X and Y be sets

sets are be a

are be a relation from X to Y you see instead of

relation from X to Y you see instead of writing it out like that I can just use

writing it out like that I can just use a notation to just compress

a notation to just compress things I'm going to say uh I'm going to

things I'm going to say uh I'm going to um say that R is

um say that R is um the graph of a function if the

um the graph of a function if the following

happens R is said to

said to be the

function so you see here I'm defining what it means for a relation to be the

what it means for a relation to be the graph of of a function I'm not still

graph of of a function I'm not still defining what it means for something to

defining what it means for something to be a function that's a different thing

be a function that's a different thing if

if for any two pairs of points for any X1

for any two pairs of points for any X1 y1 and X2

Y2 in uh

R okay recall this meant for any I'm taking two arbitrary points and this is

taking two arbitrary points and this is supposed to be

supposed to be two if the x coordinates are the same

two if the x coordinates are the same then the Y coordinates have to be the

then the Y coordinates have to be the same

same too

too so if X1 equals

so if X1 equals X2 then I want y1 to be equal to

X2 then I want y1 to be equal to Y2 that's the condition that tells you

Y2 that's the condition that tells you when a relation is a graph of a

when a relation is a graph of a function okay yeah shouldn't that work

function okay yeah shouldn't that work the other way as well no

um so for instance think of it like this it's a constant

this it's a constant function constant functions are

functions it's a relation of course but that's

that's fine now whether or not it is injective

fine now whether or not it is injective or you know one to one is a different

or you know one to one is a different matter which we're going to get

matter which we're going to get to but I think I'm out of time so we're

to but I think I'm out of time so we're going to stop here and on Friday I'm

going to stop here and on Friday I'm going to Define what it means for what

going to Define what it means for what what a function is and we're going to

what a function is and we're going to continue continue thank

you okay let's start good morning um so we have a lot to do today um we were

we have a lot to do today um we were talking about relations and then last

talking about relations and then last time I talked

time I talked about a condition um that tells you when

about a condition um that tells you when a relation is the graph of a function

a relation is the graph of a function and we're going to get to that

and we're going to get to that um which is also related to this idea of

um which is also related to this idea of constructing new sets out of old ones um

constructing new sets out of old ones um but before that I want to do some other

but before that I want to do some other things and then we're going to continue

things and then we're going to continue so first of all first problem set is up

so first of all first problem set is up on canvas um so right now if you go to

on canvas um so right now if you go to canas and if you clict the assignment

canas and if you clict the assignment page for uh problem set one there you're

page for uh problem set one there you're going to see a PDF file which you can

going to see a PDF file which you can download if you want and so on um so you

download if you want and so on um so you should read through it and um I should

should read through it and um I should mention uh something and I think I'm

mention uh something and I think I'm going to say a couple more words about

going to say a couple more words about problem sets and how you should approach

problem sets and how you should approach them what and what what what what should

them what and what what what what should be your weekly workflow and stuff like

be your weekly workflow and stuff like that but for now let's just say it like

that but for now let's just say it like this the way I designed this class is um

this the way I designed this class is um so that it's it's a cyclical thing it's

so that it's it's a cyclical thing it's an obvious thing of course but sometimes

an obvious thing of course but sometimes it's sort of like it becomes hard to

it's sort of like it becomes hard to keep track of the

keep track of the cycle okay so the cycle starts on a

cycle okay so the cycle starts on a Friday which is today let's say and then

Friday which is today let's say and then it continues right so tomorrow is

it continues right so tomorrow is Saturday and then Sunday and then Monday

Saturday and then Sunday and then Monday Tuesday

Tuesday Wednesday Thursday and then Friday next

Wednesday Thursday and then Friday next week

so and like I said before typically you're going to have

before typically you're going to have some assignment

some assignment du each Friday I e it's going to be a

du each Friday I e it's going to be a problem set or it's going to be a

problem set or it's going to be a midterm um or an exam or or a

midterm um or an exam or or a final and um for a problem set you're

final and um for a problem set you're going to

going to have basically a week to work on right

have basically a week to work on right so I uploaded problem set one today uh

so I uploaded problem set one today uh in uh later weeks I might um upload the

in uh later weeks I might um upload the problem say set later in the day but

problem say set later in the day but it's going to be certain before the day

it's going to be certain before the day ends and then you're going to have a

ends and then you're going to have a whole week

whole week and then next week on Friday at midnight

and then next week on Friday at midnight is going to be the

is going to be the deadline um now we talked about how uh

deadline um now we talked about how uh the workload for this class is 8 to 12

the workload for this class is 8 to 12 hours

but there is in fact more to that in that somehow especially when it comes to

that somehow especially when it comes to learning math having one large chunk of

learning math having one large chunk of 10 hours turns out to be not the same as

10 hours turns out to be not the same as as having multiple chunks let's say five

as having multiple chunks let's say five chunks of each of um two hours okay so I

chunks of each of um two hours okay so I would highly

would highly recommend that you at least have a look

recommend that you at least have a look at the problem set the day it is

at the problem set the day it is uploaded start tomorrow okay don't leave

uploaded start tomorrow okay don't leave it to the last

it to the last day um I'm designing the problem problem

day um I'm designing the problem problem sets uh in such a way that

sets uh in such a way that um it's going to be overwhelming if you

um it's going to be overwhelming if you try to do a in one

try to do a in one setting okay I'm designing it so that

setting okay I'm designing it so that every single day you work a little bit

every single day you work a little bit and then initially maybe things don't

and then initially maybe things don't make sense and then you keep working on

make sense and then you keep working on it let's say Saturday Sunday maybe you

it let's say Saturday Sunday maybe you start sending emails to me or you know

start sending emails to me or you know you start going to to to the Twitter

you start going to to to the Twitter Center and stuff like that and then uh

Center and stuff like that and then uh let's say on Wednesday you go to the

let's say on Wednesday you go to the tutor Center and then on Thursday you

tutor Center and then on Thursday you work on it a little bit and then on

work on it a little bit and then on Friday you finally finish the the uh

Friday you finally finish the the uh problem set and so on okay so this is

problem set and so on okay so this is very important don't leave it to

very important don't leave it to Thursday Wednesday blah blah blah start

Thursday Wednesday blah blah blah start today if not

today if not tomorrow

tomorrow okay so that's an important aspect

okay so that's an important aspect another thing

another thing um and once you spend some uh time on

um and once you spend some uh time on problem set one I think on Monday I'm

problem set one I think on Monday I'm going to say a couple more words about

going to say a couple more words about it uh

it uh so before going into these relations and

so before going into these relations and what you can do with them and functions

what you can do with them and functions and so on I wanted to mention the these

and so on I wanted to mention the these proof methods that because you're going

proof methods that because you're going to need these um and it's kind of kind

to need these um and it's kind of kind of a bizarre thing to talk about how to

of a bizarre thing to talk about how to prove things and typically it's like

prove things and typically it's like people talk about these different types

people talk about these different types of proofs but in

of proofs but in reality it's not as straightforward to

reality it's not as straightforward to distinguish this proof from that other

distinguish this proof from that other proof but at the very least I thought it

proof but at the very least I thought it would be a good idea to give you the

would be a good idea to give you the general form The Logical form of each of

general form The Logical form of each of these um argumentation types of

these um argumentation types of argumentation and then start with a very

argumentation and then start with a very um rather straightforward Claim about

um rather straightforward Claim about divisibility of numbers and sort of

divisibility of numbers and sort of prove that statement using every single

prove that statement using every single one of them so that we have an idea as

one of them so that we have an idea as to how this would work but before

to how this would work but before continuing are there any

questions by the way that uh seat might be not a good seat because you cannot

be not a good seat because you cannot see me here I mean you can see me here

see me here I mean you can see me here but not this stuff here unfortunately

but not this stuff here unfortunately this it's the I thought about it I

this it's the I thought about it I thought about whether or not uh I should

thought about whether or not uh I should just not write things here but we have

just not write things here but we have so little space that I'm just going to

so little space that I'm just going to blame the design of the room okay they

blame the design of the room okay they should have put the uh um board over

should have put the uh um board over there I think instead of

there I think instead of here anyway okay so the statement I want

here anyway okay so the statement I want to think about is the following I'm just

to think about is the following I'm just going to call it claim okay there's a

going to call it claim okay there's a bunch of of sort of words in math uh

bunch of of sort of words in math uh speak let's say that sort of like

speak let's say that sort of like signify something that is claimed to be

signify something that is claimed to be proven and something like that you could

proven and something like that you could see claim proof corollary LMA

see claim proof corollary LMA proposition blah blah blah result these

proposition blah blah blah result these are all like words I'm just going to

are all like words I'm just going to leave it as claim for now there's a

leave it as claim for now there's a certain rhetoric going on of course when

certain rhetoric going on of course when you call the fundamental theorem of

you call the fundamental theorem of calculus the fundamental theorem of

calculus the fundamental theorem of calculus instead of the fundamental

calculus instead of the fundamental claim of calculus okay claim is the

following um let N be an integer it's a number the claim is if n

integer it's a number the claim is if n is odd then 3 n + 7 is even

integer if n is odd then 3 n + 7

odd then 3 n + 7 is

is even so that's the

even so that's the claim nothing too crazy about it of

claim nothing too crazy about it of course you have an odd number number you

course you have an odd number number you multiply it by another odd number that's

multiply it by another odd number that's again odd and then you add yet another

again odd and then you add yet another odd number that gives you an even number

odd number that gives you an even number fine but um the point of this uh thing

fine but um the point of this uh thing is to give sort of like some um

is to give sort of like some um exposition of these things these proof

exposition of these things these proof methods um do I want to say anything

methods um do I want to say anything before that oh let's do one thing let's

before that oh let's do one thing let's try

try to turn this into logic

to turn this into logic so let's try to sort of use the logical

so let's try to sort of use the logical symbol symbols and the set theory

symbol symbols and the set theory symbols I'm going to write it in two

symbols I'm going to write it in two different ways they that are

different ways they that are equivalent

equivalent um AKA using logic well AKA let me say

um AKA using logic well AKA let me say it like this um so I can write it like

it like this um so I can write it like so for any n in

Z if any is odd so how can I represent n being odd well n being odd is is the

being odd well n being odd is is the same thing as n is not even in other

same thing as n is not even in other words it is not divisible by two well

words it is not divisible by two well the notation for this uh turns out to be

the notation for this uh turns out to be the

the following so this represents two divides

following so this represents two divides n and you put a dash here that

n and you put a dash here that represents not so two doesn't divide

n implies two divides

implies two divides 3 n + 7 so I can represent the same

3 n + 7 so I can represent the same statement like

statement like this any questions about

this using set theory I can instead of sort of using these divisibility symbols

sort of using these divisibility symbols I can use

I can use sets um and write it like

sets um and write it like so again for any

so again for any n in Z any

integer if n is not in 2 Z it's a set well what is

n is not in 2 Z it's a set well what is the set it it's the set that represents

the set it it's the set that represents even

even numbers so maybe I should write

if n is not there then 3 n + 7 is in 2

Z okay I can write it like this as well so these are all the same

so these are all the same statements

okay and in problem set one two you're going to see some you're going to see

going to see some you're going to see some parts of certain problems where I

some parts of certain problems where I describe something in using in in plain

describe something in using in in plain English and I ask you to rewrite it or

English and I ask you to rewrite it or expressit using set theory and then

expressit using set theory and then there's going to be some set theoretical

there's going to be some set theoretical sort of equation and I'm going to there

sort of equation and I'm going to there I'm asking you to sort of rewrite it or

I'm asking you to sort of rewrite it or um restate it using plain

um restate it using plain English

English Okay questions about this so far okay

Okay questions about this so far okay direct

direct proof is of the following for

proof is of the following for you want to prove a statement of the

you want to prove a statement of the form P implies

form P implies Q now in this case really uh there are

Q now in this case really uh there are like quantifiers and stuff like that but

like quantifiers and stuff like that but let's not get into the sort of like the

let's not get into the sort of like the specifics of this so you want to prove

specifics of this so you want to prove this maybe I

this maybe I should um

want to prove for each of them let's say you

prove for each of them let's say you want to prove P implies

want to prove P implies Q for two

statements for Drake proof assume P you don't know if p is true necessarily but

don't know if p is true necessarily but you assume it and then through some

you assume it and then through some logical manipulations and some math

logical manipulations and some math stuff you deduce Q okay that's why it's

stuff you deduce Q okay that's why it's called direct proof

plus logic plus

logic plus math somehow gives you Q that's the

math somehow gives you Q that's the general type General sort of structure

general type General sort of structure of a direct

of a direct proof so here is how I would prove this

proof so here is how I would prove this using a direct proof

now I'm going to assume the Assumption and I'm going to try to come up with

this um let me say this as well um I'm going to try to be rather strict

um I'm going to try to be rather strict when I'm

when I'm writing these specific examples but as

writing these specific examples but as the course progresses I'm going to be

the course progresses I'm going to be more and more loose and I'm going to

more and more loose and I'm going to sort of leave certain parts of proofs to

sort of leave certain parts of proofs to you to

you to finish

finish okay let N

okay let N be an

be an integer and

suppose um it is not so it's it's not divisible by

divisible by two what does this mean this means that

two what does this mean this means that it's odd in other words there is some

it's odd in other words there is some integer K such that n equal 2K + 1

integer K such that n equal 2K + 1 that's literally what it means for a

that's literally what it means for a number to be an odd

number to be an odd number then there is a k

1 so when you divide n by two you have a remainder and the only remainder that

remainder and the only remainder that can be is one in this case of course

can be is one in this case of course um now I need to figure out something

um now I need to figure out something about this number then

3 n + 7 well I can just substitute n = 2K + 1 for n

here and then consolidating what do I get 6 k + 3 + 7 is 6 k +

10 and then I can factor out a two because this is equal to 2 * 3 k +

5 now K is an integer you see an integer time an integer is an

see an integer time an integer is an integer the sum of two integers is also

integer the sum of two integers is also an integer which means that 3K + 5 is an

an integer which means that 3K + 5 is an integer this guy

in other words this uh chain of equalities really tells me that this

equalities really tells me that this number is twice some some integer that

number is twice some some integer that is by Def that that is a definition of

is by Def that that is a definition of this being

even okay and that's the end of the proof now typically you can just leave

proof now typically you can just leave it like this but it's sort of like good

it like this but it's sort of like good form to sort of sign ify that you

form to sort of sign ify that you finished the proof um so one thing

finished the proof um so one thing people do is they put sort of like a

people do is they put sort of like a square and stuff like

square and stuff like that some people would like to write

that some people would like to write like

like Q it's it stands for something Latin

Q it's it stands for something Latin which basically says that the proof is

which basically says that the proof is over uh there's a bunch of other things

over uh there's a bunch of other things you can do like IMT it's Miller time

you can do like IMT it's Miller time uh i' I've seen GG I've seen uh rip Boo

uh i' I've seen GG I've seen uh rip Boo and other things there's a bunch of

and other things there's a bunch of things Okay so

things Okay so so you're just learning these kind of

so you're just learning these kind of things so you might as well just get it

things so you might as well just get it out of your system obviously these are

out of your system obviously these are these are not professional things

these are not professional things but when you're young when you're first

but when you're young when you're first learning these kind of things you're

learning these kind of things you're excused okay any questions about

excused okay any questions about this obviously in this case things were

this obviously in this case things were rather straightforward as we go further

rather straightforward as we go further and further proving things are going to

and further proving things are going to be more and more complicated I chose a

be more and more complicated I chose a statement because pretty much you don't

statement because pretty much you don't need any calculus or anything anything

need any calculus or anything anything advaned to go through

advaned to go through this so for instance proving that root2

this so for instance proving that root2 is uh not a rational number is slightly

is uh not a rational number is slightly more complicated of a proof still it's

more complicated of a proof still it's not super complicated still but you you

not super complicated still but you you need something compared to this okay so

need something compared to this okay so this is direct proof proof by Contra

this is direct proof proof by Contra positive is the following so again you

positive is the following so again you want to prove P implies Q now instead of

want to prove P implies Q now instead of starting with P you're going to you're

starting with P you're going to you're going to assume not Q okay now let me

going to assume not Q okay now let me not say too much about this but it turns

not say too much about this but it turns out that I think I mentioned this last

out that I think I mentioned this last time P implies Q turns out to be

time P implies Q turns out to be logically equivalent to not P not Q

logically equivalent to not P not Q implies not P so you want to assume not

implies not P so you want to assume not q and then plus logic and then plus math

q and then plus logic and then plus math and then you want to show not

and then you want to show not Q so let me write it so for this

Q not Q um well maybe I should write it like

Q um well maybe I should write it like this so this was my symbol for not the

this so this was my symbol for not the complement

um plus logic plus math

logic plus math and then hopefully you're going to get

and then hopefully you're going to get not p as a

not p as a conclusion

okay and these uh methods also have Latin names and stuff like that

Latin names and stuff like that obviously you don't need to know what

obviously you don't need to know what they are called but you need to be able

they are called but you need to be able to do you need to be able to perform

to do you need to be able to perform them okay so let me just uh make some

them okay so let me just uh make some room here and I'm going to delete these

room here and I'm going to delete these uh logical statements as well notation

uh logical statements as well notation statements as

well I'm going to I'm going to take the same claim but the uh structure of my

same claim but the uh structure of my proof is going to be different this

time so now I want to prove the claim using this proof by Contra positive

okay so now I need to assume not this so suppose so let N be an integer and

suppose so let N be an integer and suppose that 3 n + 7 is not even right

suppose that 3 n + 7 is not even right I'm just making I'm just negating the

I'm just making I'm just negating the statement well not even this is is the

statement well not even this is is the same thing as odd

same thing as odd so let

concise suppose um 3

suppose um 3 n + 7 is not an element right so element

n + 7 is not an element right so element symbol but I put a dash that means not

in to it's not an so suppose that it's not an

it's not an so suppose that it's not an even

even number thus again I'm going to use the

number thus again I'm going to use the same trick um there's a number

K such that 3 n + 7

that 3 n + 7 equals 2K +

here thus um oh let's do this I don't want to

um oh let's do this I don't want to divide by three here for some reason

divide by three here for some reason you're going to see in a second but I

you're going to see in a second but I can send the seven to the other

can send the seven to the other side thus 3 n

equals 2K + 1 - 7 and of course as you're writing proofs and stuff like

you're writing proofs and stuff like that you don't need to sort of keep

that you don't need to sort of keep writing sort of these intermediary steps

writing sort of these intermediary steps uh but just because this is the first

uh but just because this is the first time we're writing proofs I'm sort of

time we're writing proofs I'm sort of being uh more explicit of course this is

being uh more explicit of course this is the same thing as 2K minus 6 and then I

the same thing as 2K minus 6 and then I can factor out to

two now 3 and then is an even number right because K

then is an even number right because K minus 6 is an integer I'm saying that 3

minus 6 is an integer I'm saying that 3 n is an even number three say

n is an even number three say again uh yeah thank

again uh yeah thank you okay so I factored out a two here

you okay so I factored out a two here and as your colleague suggests this

and as your colleague suggests this should be a three in any event the

should be a three in any event the argument still holds K minus 3 is an

argument still holds K minus 3 is an integer and I'm saying here uh that 3 n

integer and I'm saying here uh that 3 n this integer is an even

this integer is an even number well

number well If the product of two

If the product of two numbers um is divisible by two well at

numbers um is divisible by two well at least one of them has to be that

least one of them has to be that way which kind of looks like a gap here

way which kind of looks like a gap here which in a certain sense it is but uh

which in a certain sense it is but uh let's just hold on to that

let's just hold on to that um maybe I can write it like so since 3

um maybe I can write it like so since 3 is uh

is uh prime it is not divisible by anything

prime it is not divisible by anything except one and three it must be the case

except one and three it must be the case that two is divisible by two uh n is

that two is divisible by two uh n is divisible by

two it must be the

in other words N is even uh what did I do okay that's Co

even uh what did I do okay that's Co good in other

words N is let me say not odd so I assumed

odd so I assumed um the negation of the conclusion and I

um the negation of the conclusion and I was able to end up with the ation of the

was able to end up with the ation of the hypothesis and so that's the proof by

hypothesis and so that's the proof by Contra

positive any question about this now I mentioned something about

this now I mentioned something about there being perhaps a gap in this

there being perhaps a gap in this step

step um probably for this class we can just

um probably for this class we can just say this we can assume

say this we can assume okay

okay but if you wanted to prove it any

but if you wanted to prove it any guesses as to how you would sort of

guesses as to how you would sort of justify this step this sentence here so

justify this step this sentence here so I have two numbers I'm saying if the

I have two numbers I'm saying if the product is even at least one of them has

product is even at least one of them has to be

even you have to prove that like two divides to

divides to a yes ultimately that's the statement

a yes ultimately that's the statement I'm using and it turns out that there's

I'm using and it turns out that there's a certain fundamental theorem not of

a certain fundamental theorem not of calculus of course but of what is called

calculus of course but of what is called arithmetic which says that any integer

arithmetic which says that any integer has a unique prime

has a unique prime factorization and if you take an

factorization and if you take an elementary uh number Theory class blah

elementary uh number Theory class blah blah blah there you would be interested

blah blah there you would be interested in these kind of proofs but for this

in these kind of proofs but for this class let's just leave it as is

class let's just leave it as is okay

okay so so there's a possible Gap here

and this is really one of the reasons why talking about proofs like this is

why talking about proofs like this is kind of

kind of iffy um because we're not sort of trying

iffy um because we're not sort of trying to build all of math from the from from

to build all of math from the from from nothing basically we're just sort of

nothing basically we're just sort of like assuming a bunch of things so that

like assuming a bunch of things so that we can start learning calculus when

we can start learning calculus when start proving things in

start proving things in calculus

calculus um but even in general you know when

um but even in general you know when you're uh practicing math typically for

you're uh practicing math typically for instance even if you're doing pure math

instance even if you're doing pure math it turns out that there's no such thing

it turns out that there's no such thing as 100% complete proof okay what it is

as 100% complete proof okay what it is is that when we're talking about proofs

is that when we're talking about proofs and when we're talking about trigger in

and when we're talking about trigger in uh math we're really talking about

uh math we're really talking about arguments that can be in principle

arguments that can be in principle completed okay there's always some

completed okay there's always some amount of Gap so when we're talking

amount of Gap so when we're talking about oh you took calculus before and

about oh you took calculus before and you know how to compute things and you

you know how to compute things and you you know certain definitions and now

you know certain definitions and now we're going to do things more rigorously

we're going to do things more rigorously we're really talking about something uh

we're really talking about something uh comparative we're going to be more

comparative we're going to be more rigorous compared to what you did before

rigorous compared to what you did before but not 100% rigorous

but not 100% rigorous okay and of course just because I'm

okay and of course just because I'm leaving this as a gap you see there's a

leaving this as a gap you see there's a difference even though I'm I'm sort of

difference even though I'm I'm sort of like recognizing that I'm oh maybe

like recognizing that I'm oh maybe there's something here that I would need

there's something here that I would need I'm not really going into it so um it's

I'm not really going into it so um it's going to be a good thing to be able to

going to be a good thing to be able to distinguish oh where is the gap if at

distinguish oh where is the gap if at all and if there is a gap and if someone

all and if there is a gap and if someone asks me to prove it how do I answer

asks me to prove it how do I answer it

it okay any questions about

okay any questions about this and here's an interesting thing too

this and here's an interesting thing too this first method is logically

this first method is logically equivalent to the second method okay but

equivalent to the second method okay but somehow as you go through it you see

somehow as you go through it you see mathematically the second Pro proof is

mathematically the second Pro proof is more complicated you need more things to

more complicated you need more things to do the to um actuate the second method

do the to um actuate the second method instead of the first method and this is

instead of the first method and this is sort of one of the mysteries of math um

sort of one of the mysteries of math um it turns out that even though a bunch of

it turns out that even though a bunch of things are logically equivalent because

things are logically equivalent because ultimately if something is is is a

ultimately if something is is is a theorem it is is a it's a toy is what a

theorem it is is a it's a toy is what a logician would say somehow it turns out

logician would say somehow it turns out that

that um for for certain statements applying

um for for certain statements applying this

this argument is much easier uh compared to

argument is much easier uh compared to this other argument even though they are

this other argument even though they are ultimately equivalent of course you're

ultimately equivalent of course you're proving the same

proving the same thing okay so that's the second method

thing okay so that's the second method third method is proof by contradiction

third method is proof by contradiction and in a sense I would say this is the

and in a sense I would say this is the most

most um this is the method where you have the

um this is the method where you have the most amount of stuff to deal with to to

most amount of stuff to deal with to to to to to play with

to to to play with and this is also perhaps some uh sort of

and this is also perhaps some uh sort of logical systems don't really accept this

logical systems don't really accept this method so it really is slightly easier

method so it really is slightly easier compared to the other

compared to the other two and so again if I want to prove P

two and so again if I want to prove P implies Q what I'm going to do here to

implies Q what I'm going to do here to prove it using contradiction is the

prove it using contradiction is the following I'm going to assume both p and

following I'm going to assume both p and not q and then I'm going to try to come

not q and then I'm going to try to come up with something that is blatantly

up with something that is blatantly false

assume p and not

p and not q and then plus logic plus

math will hopefully lead you to something blatantly false

something like 1 equals z or some other thing okay and again it turns out that

thing okay and again it turns out that from a logical point of view the

from a logical point of view the statement P implies Q is equivalent to

statement P implies Q is equivalent to the

the statement

statement uh to sorry P implies

uh to sorry P implies Q is equivalent to the negation of

Q is equivalent to the negation of this so you're really assuming you want

this so you're really assuming you want to assume is you want to prove P implies

to assume is you want to prove P implies Q you say say this doesn't work and then

Q you say say this doesn't work and then you try to come up with certain things

you try to come up with certain things okay but for now let's just leave it

okay but for now let's just leave it like

like this so proof by

this so proof by contradiction same statement

contradiction same statement um goes like this

so I'm going to assume both things both that n is odd and that 3 n + 7 is not

that n is odd and that 3 n + 7 is not even okay

let N be an

suppose and if you want to sort of signify that uh you're using this method

signify that uh you're using this method of contradiction you can say something

of contradiction you can say something like for a contradiction here but I'm

like for a contradiction here but I'm just going to not write it like that

just going to not write it like that suppose n is

suppose n is odd uh

even AKA that it is Odd as well let me not write

well let me not write it

it okay thus well I'm just going to use the

okay thus well I'm just going to use the same trick you see ultimately it's the

same trick you see ultimately it's the same idea I'm just taking this notion of

same idea I'm just taking this notion of being an odd or even integer and turning

being an odd or even integer and turning it into some expression some

it into some expression some mathematical expression that I can

mathematical expression that I can manipulate and do some algebra with thus

manipulate and do some algebra with thus there are some numbers K and

L such that n equals 2K + 1

1 and 3 n + 7 equals to

and 3 n + 7 equals to l+ one now here I need to choose two

l+ one now here I need to choose two distinct in possibly distinct integers

distinct in possibly distinct integers at the very least I should be using two

at the very least I should be using two different variables because quite

different variables because quite possibly this K is not the same thing as

possibly this K is not the same thing as this L I'm not saying I shouldn't be

this L I'm not saying I shouldn't be assuming I should I shouldn't be

assuming I should I shouldn't be claiming that n equal 3 n + 7 here there

claiming that n equal 3 n + 7 here there are possibly different things that's why

are possibly different things that's why I'm using different letters here and so

I'm using different letters here and so what do I need what what can I do here I

what do I need what what can I do here I can just substitute n = 2 k + 1 in this

can just substitute n = 2 k + 1 in this equation I have two equations well if

equation I have two equations well if you want these are linear equations if

you want these are linear equations if you know and you can solve it

you know and you can solve it right thus

2 L + 1

L + 1 equal 3 n + 7 and then substituting I

equal 3 n + 7 and then substituting I have 3 * 2 k + 1 uh + 7 which then

have 3 * 2 k + 1 uh + 7 which then consolidating gives me 6 k + 3 + 7 which

consolidating gives me 6 k + 3 + 7 which we had before already uh which is 6 k +

we had before already uh which is 6 k + 10

okay what to do here so I'm trying to come up with something wrong okay

come up with something wrong okay something has

something has to uh something has to break well one

to uh something has to break well one thing I can do is the following I can

thing I can do is the following I can solve I can I can collect the constant

solve I can I can collect the constant terms on one side and the variable terms

terms on one side and the variable terms on the other side so this would this

on the other side so this would this would give

would give me the following so 2 L minus 6K oh

me the following so 2 L minus 6K oh maybe I should write it the other way

maybe I should write it the other way around so I'm going to send one to the

around so I'm going to send one to the other side I have 9 here and then this

other side I have 9 here and then this 6K to the other side over there um

6K to the other side over there um that's 9

that's 9 equal 2 l - 6 K which is the same thing

equal 2 l - 6 K which is the same thing as 2 * l - 3K you see it's the same

as 2 * l - 3K you see it's the same trick this guy is an integer so that

trick this guy is an integer so that this really says that 9 is an even

this really says that 9 is an even number which is obviously true obviously

number which is obviously true obviously not true

but nine is not even and I like to sort of State

even and I like to sort of State something like a contradiction at the

something like a contradiction at the end of a proof by contradiction at the

end of a proof by contradiction at the very least I want to say something

very least I want to say something around something in the proof that you

around something in the proof that you know I'm sign I'm I want to signify that

know I'm sign I'm I want to signify that I'm using a proof by

I'm using a proof by contradiction a

so ultimately you see all three of these arguments essentially is based on the

arguments essentially is based on the same mathematical idea you basically

same mathematical idea you basically write down the definition of what it

write down the definition of what it means for an integer to be odd or even

means for an integer to be odd or even and then you sort of manipulate that's

and then you sort of manipulate that's sort of like the high level picture but

sort of like the high level picture but when it comes to the implementation the

when it comes to the implementation the implementation the details could be

implementation the details could be could be different

um so here

here okay so do you agree that 2 L + 1 equal

okay so do you agree that 2 L + 1 equal 6K + 10 okay so I can send to the other

6K + 10 okay so I can send to the other side 10 - 1 is

9 any other questions yes could we also have divided the six PL sorry K 10 into

have divided the six PL sorry K 10 into 2 * 3 + 5 yeah there are infinitely yes

2 * 3 + 5 yeah there are infinitely yes there are infinitely many variations of

there are infinitely many variations of these things

cool so that's it these are the three methods there are a couple more well the

methods there are a couple more well the more important one that we're going to

more important one that we're going to spend some time on uh is what is called

spend some time on uh is what is called proof by induction when you have sort of

proof by induction when you have sort of like a statement that is indexed by

like a statement that is indexed by integers let's

integers let's say for instance uh the sum of integers

say for instance uh the sum of integers from one to n or something like that

from one to n or something like that there it becomes useful to use this

there it becomes useful to use this thing called integers uh induction but

thing called integers uh induction but um these are sort of like the uh the

um these are sort of like the uh the methods that that's going to be helpful

methods that that's going to be helpful for this week's uh problem set yes

for this week's uh problem set yes um after I those statement and then I

um after I those statement and then I just

just 2 then do the same stuff but but when I

2 then do the same stuff but but when I got this uh like 3 plus 7 isal 6 k + 10

got this uh like 3 plus 7 isal 6 k + 10 then I factor that 2 3 + 5 then it's a

then I factor that 2 3 + 5 then it's a it's a even number it's not all so

it's a even number it's not all so there's a contradiction can do that is

there's a contradiction can do that is that correct or well I mean in so far as

that correct or well I mean in so far as you get a contradiction it is correct

you get a contradiction it is correct but okay so here's the thing though yeah

but okay so here's the thing though yeah yeah

yeah do so ultimately do you get a

do so ultimately do you get a contradiction ultimately do you get a

contradiction ultimately do you get a contradiction in that yeah okay then yes

contradiction in that yeah okay then yes oh see

yes yes so if we do something like this on the homework problem yes would that

on the homework problem yes would that be full marks why shouldn't

be no no it's good it's good just just check

questions okay um I want to say a couple more words about this uh about relations

more words about this uh about relations I want to discuss functions and stuff

I want to discuss functions and stuff like that one thing I wanted to do last

like that one thing I wanted to do last time that I forgot is um a bunch of

time that I forgot is um a bunch of operations that you can do with

operations that you can do with relations uh let's discuss those first

relations uh let's discuss those first and then we're going to finalize with

and then we're going to finalize with this notion of of a function

relations now these um are described in the textbook

um are described in the textbook specifically for functions and in a

specifically for functions and in a second we're going to see how functions

second we're going to see how functions are really nothing but spe special types

are really nothing but spe special types of function um special types of

of function um special types of relations but all of these can be

relations but all of these can be defined more generally for relations

defined more generally for relations what I'm not going to do is I'm not

what I'm not going to do is I'm not going to give examples it's going to be

going to give examples it's going to be your job okay and this is going to be a

your job okay and this is going to be a standing standing

standing standing exercise um whereas in a sort of um

exercise um whereas in a sort of um calculation based um um math class let's

calculation based um um math class let's say what is important is in class is to

say what is important is in class is to have a bunch of examples and stuff like

have a bunch of examples and stuff like that here it's going to be a little

that here it's going to be a little different building examples is going to

different building examples is going to be part of your job like I discussed on

be part of your job like I discussed on Wednesday that's one of you one of the

Wednesday that's one of you one of the methods in which you approach

methods in which you approach problems okay so I want to have three

this one relation R from X to Y and then yet another relation let's call it s

yet another relation let's call it s from y to Y to

Z weall the notation here x * y was the cartisian product it's the set of all

cartisian product it's the set of all those ordered pairs X comma y lowercase

those ordered pairs X comma y lowercase x comma lowercase y such that lowercase

x comma lowercase y such that lowercase x comes from uppercase X and lowercase y

x comes from uppercase X and lowercase y comes from uppercase Y and then this was

comes from uppercase Y and then this was the set of all um subsets it was a power

the set of all um subsets it was a power set I'm just saying this R is a subset

set I'm just saying this R is a subset of x * Y and likewise for S and let me

of x * Y and likewise for S and let me also take a subset of

also take a subset of x a

x a in P of X okay now I'm going to define a

in P of X okay now I'm going to define a bunch of things yes should the subs

bunch of things yes should the subs symbol being used in of

symbol being used in of no okay good

this when you put the P here you go to The

The Meta okay so you it allows you to sort

Meta okay so you it allows you to sort of replace this expression with this

expression this same thing and in fact in problem set one

thing and in fact in problem set one you're going to see

you're going to see uh something like

uh something like this okay part of it is because I want

this okay part of it is because I want you to think step by step and sort of

you to think step by step and sort of unpack the definition and sort of repack

unpack the definition and sort of repack them and all that

them and all that stuff

stuff okay so I need the following things

okay so I need the following things domain image inversion composition

domain image inversion composition restriction and image of a

restriction and image of a subset so

domain these are definitions all this is a certain subset

definitions all this is a certain subset of

of X and I'm going to write it like so Dom

X and I'm going to write it like so Dom of R is the set of all those x's in X

of R is the set of all those x's in X such that X such that R relates X to sum

such that X such that R relates X to sum y

okay image or range of R is sort of like the opposite

thing set of all those y's such that for some X

some X X is related to

Y I understand that it's it's a barrage of notation and definitions and stuff

of notation and definitions and stuff like that

like that um hopefully throughout the semester

um hopefully throughout the semester these are going to be more and more um

these are going to be more and more um easier to

easier to parse these are not super important for

parse these are not super important for us that's why I'm sort of fast

us that's why I'm sort of fast forwarding it I think I am uh legal

forwarding it I think I am uh legal obliged to mention something about these

obliged to mention something about these so

inversion or so let's say inverse inverse of

one and this is uh the following it's a set of all those y comma X's such that X

set of all those y comma X's such that X comma Y is in r

this is sort of like a transpose if you want if you're familiar with matrices

want if you're familiar with matrices and

and stuff next up composition and so this is

stuff next up composition and so this is where I start using this other

so s composed with r the notation for this is this circle small circle well

this is this circle small circle well it's a set of all those pairs X comma

it's a set of all those pairs X comma Z in x * Z such that for some y x is

Z in x * Z such that for some y x is related to y y r and Y is related to z y

related to y y r and Y is related to z y s

so now I want to use this subset a the Restriction of R to a I'm going to

a the Restriction of R to a I'm going to denote by R and then a bar Long Bar and

denote by R and then a bar Long Bar and then I'm going to put a it's almost like

then I'm going to put a it's almost like r evaluated at a in terms of like

r evaluated at a in terms of like integrals and stuff and so this is

integrals and stuff and so this is nothing but X comma Y in R such that I'm

nothing but X comma Y in R such that I'm restricting the input value to be coming

restricting the input value to be coming from

from a and then finally the image of a under

a and then finally the image of a under R is the following

this is denoted by R of a it's almost like a function and in fact uh part of

like a function and in fact uh part of your uh so it's going to be a part of

your uh so it's going to be a part of problem set one to really make sense of

problem set one to really make sense of sense of this expression as a function

sense of this expression as a function now this is nothing but the set of all

now this is nothing but the set of all those y's such that

those y's such that um there's an a in a such

um there's an a in a such that a comma Y is in r

this really is the same thing as the image of the Restriction so I can

as the image of the Restriction so I can you can write it like this as

well exercise is to come up with visual representations visual examples for all

representations visual examples for all of these and I would suggest just take X

of these and I would suggest just take X Y and Z to be let's say the unit

actually even when I'm not stating this it's always you should always take it as

it's always you should always take it as an exercise to come up with

I realize this is too much too many definitions without uh justification and

definitions without uh justification and stuff let's just say that uh we're going

stuff let's just say that uh we're going to we're going to use these things uh

to we're going to use these things uh sporadically speaking let's

sporadically speaking let's say it's just a bunch of terminology and

say it's just a bunch of terminology and notation um and in the future if we end

notation um and in the future if we end up needing to use things we're going to

up needing to use things we're going to sort of I'm going to remind you these

sort of I'm going to remind you these things okay we have a couple minutes um

things okay we have a couple minutes um let me just finish this relation this um

let me just finish this relation this um definition of a

function so this was related to the Third Way of defining or constructing a

Third Way of defining or constructing a new new set out of old

ones recall that what he said was um that for r a relation it is the graph of

that for r a relation it is the graph of a function

for any two points in R if X1 equal X2 then y1 has to be equal to

then y1 has to be equal to Y2 in this case I'm going to put a to be

Y2 in this case I'm going to put a to be equal to the domain of r as I just

equal to the domain of r as I just defined

defined um I think I'm running out of time

um I think I'm running out of time actually again

actually again this is cursed I'm still I still

this is cursed I'm still I still couldn't finish this definition we're

couldn't finish this definition we're going to continue this on Monday thank

going to continue this on Monday thank you have a nice weekend I'll see you on

you have a nice weekend I'll see you on Monday

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