Question

Set Theory

Answer 1

\(\large \color{black}{\begin{align} \normalsize \text{A ∩ B = A ∩ C need not imply B = C.} \quad \normalsize \text{ Explain through an example.}\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

A = {1}
B = {1, 2}
C = {1, 3}

Answer 3

huh kai was so fast :P

Answer 4

Hahaha there could have been a simpler example:

A = {}
B = {1}
C = {2}

Answer 5

thnx

Answer 6

I think that last example I gave sorta shows that it's almost like this is a lot like multiplying by zero.

a*b=a*c

This doesn't mean b=c, since a=0 is possible.

Answer 7

Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set
X, show that A = B.

Answer 8

in general any case with \[ A\cap B \neq \varnothing \] does not imply A=C

Answer 9

or \(C\cap B \neq \varnothing \)

Answer 10

i need an example like the previous one

Answer 11

ikram needs more owl bucks sry

Answer 12

A={1,2,3}
B={1,2,3}
C={1,2,3,4}

Answer 13

:O @Kainui i dont lol

Answer 14

Hahaha jk :P

Answer 15

is that the example for this que

Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set
X, show that A = B.

Answer 16

you wanna prove this ?

Answer 17

yes with example

Answer 18

ok example
A={1,2,3}
B={1.4}
C={1,2.3}

Answer 19

but C is not there in the question

Answer 20

sorry i ment to say x :P

Answer 21

now to prove they are two side of the prove
1- show A is a subset of B
2- show B is a subset of A

Answer 22

let \(c\in A \cup X\) so \( c\in A ~or~c\in X \) if c in A then
\( c\in B\cup X \) and \( c\in B\) (since c is not in x) thus \(A\subset B\)
the other way is alike :P sorry im lazy to type

Answer 23

Is it possible to do something like this?

\[
A\cup X=B\cup X\land A\cap X=B\cap X=\emptyset\implies A\cup X\setminus X=B\cup X\setminus X\implies A=B
\]