This content appears to be a transcript of a lecture or tutorial explaining fundamental concepts in set theory and functions, including set notation, operations, probability, and different types of functions.
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so let's
function simple examp
so X belongs to n such that X is less
N means natural
numbers natural numbers numberal
8 Simple simple
number R
1 2 3 4 5 6 7 8 9
2 5 into
5 but 5 into 9 5 x so 5 into
9 45 45
37 accepted
xal to
10 5 into 10
10
10
a x such that X belongs to U comma 5x is
greater than
good
sare
Pary all right REM
toal number X is such that a X belongs
okay a set a
two B [Applause]
B belongs to
one
right
2 Cube
8 2 the^ 4 16 2 power 5
question AAL X belongs to n
n 4 is less than
20 x value
one
two one
two 1
okay
Bal p a p of B P of a intersection
B P of a union P of
one p of beta
2 2 5
of P of
bet a b
p a
a intersection
a p of a intersection
P of
a intersection P of
for 2
64 so 64 answer
Rak by uh Char Char
a e
e
er no
so shanto Manan fixed
fixed
16 5 25 36 49 64 81 100 12
[Music] sa
set so a set
set 1
1 4
4 16
2 25 3 six Dot
set onto t
set bage set
b a prime set a prime intersection B Prime just
PB PB man B set sh set Union C set sh set P Union
set P Union b b sorry B
13 so 2 to the power 2 4 2 power
4 2 power form PC
so 0 common one common three
to say and ultimately 1315
c b un
c b un c 1 3 or five
so P of B un
belongs to belongs to P
[Music] PC C
3 13 p b un3
one five done
135 PB Union
PB Union PC belongs to p b Union
PC belongs to p b Union c pro
49 number of G
section value number of intersection H
value number of intersection H intersection B intersection G
number of a plus number of B plus number of C minus number of a union
of C minus number of a union b b Union C
number a intersection b b intersection
okay first of all number of number of B number of C
B three number
three number of Gap
of Gap belly Gap
Bell so number of Gap sorry Gap h
B NG minus
NG minus intersection intersection key h
intersection intersection key h g b
g b [Music]
intersection and then
number of a plus number of B it number of
that and simple okay
number of cricket number of
so a p of a intersection B
4 second first bracket 2 three 2
[Music] oh function
Ys y = root 4- x² [Music]
X such that X belongs to z z Square value less oral to four
value less oral to four Z right
s- 2-1 0 1 2 plus- one plusus
s y = root over 4 -
over 4 - x² 4 minus x
x² 4 minus x square short
X belongs to a or Y belongs to a comma X Val
2 x Val plus one
root3 Val Z
root3 not belongs to root3 root3 not belongs to
S - 2 0
0 02 2
0us 2 0 plus 2
2 domain domain
domain domain set minus 2
0+ dom dom dom do - 2 0 +
belongs to3
- 2 - 1 0 1 2 - 2
2 - 2 -1 0 1 2us
-1 0 1 2us 2us 3- one Inus 2
2us 3- one Inus 2 -1 1
minus one 01 2 domain minus 2 minus one 01 range
function function similar sort of things but main
difference Ty
like so special relation so special Rel
function f Dom
same input sorry
now possible but different output
[Music] possible
= so Yus
function f y 2x F
x + 1 - 1 2x - sorry + 2 -
1 2x - sorry + 2 - 1
1 2x-
2x- 1 x +
1 F half function f of x main
function so 2 into halfus
one 1- 1 is z f of half value zero and f x + 1 value 2x - 1
9 four three 9 or8
three 9 or8 ag8 4 3 9
x minus X say 2 by 2 minus x a function
[Music] pluss
pluss to condition X is not equal to 2
Beal 10 should be
minus [Music]
f f
f f x belongs to
1 2x should be not equal to - 3 x should be not equal to minus
okay function
function function function onto onto
x y - y 2x + 2
y - 2 y + 2
2 x y + 2 y -
x y + 2 y - 2 done
range y belongs to R such that Y is not equal to
Y is not equal to 2 but Y is not equal to
done topic function
inut X1 F of
X1 F of X1 X1 X2
F of X1 F of X2 input this output so X1
different X1 or X2 different different do X1 or X2
[Music] domain f of
over X1 - 4 F of
4 F of X2
X2 root X2
X1 F of X1 root X1 minus 4 X2 X2
4 X1 - 4 X2 - 4 so 4 4 X1
X2 function range same domain same input
function just could be easy simple
X12 [Music]
Square that's on function on
on function on function function
Y 2us
2us 2 - 5 - 3 y
+ y - 2 y +
2 y + 2
2 x y + 2
simple x + 2 x- 2
inverse inverse x + 2 x -
inverse P Val KP KP p+ P
minus KP squ minus 2 KP p
2 p +
[Music] f
so yis so
y² x - 4 y² + 4 so F of inverse X
4 y² + 4 so F of inverse X function x² + 4
output valus 28
2 y x²
y x² yal
yal root
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