## mathmath333 one year ago Set Theory

1. mathmath333

\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}}

2. Kainui

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

3. ikram002p

huh kai was so fast :P

4. Kainui

Hahaha there could have been a simpler example: A = {} B = {1} C = {2}

5. mathmath333

thnx

6. Kainui

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.

7. mathmath333

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.

8. ikram002p

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

9. ikram002p

or $$C\cap B \neq \varnothing$$

10. mathmath333

i need an example like the previous one

11. Kainui

ikram needs more owl bucks sry

12. ikram002p

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

13. ikram002p

:O @Kainui i dont lol

14. Kainui

Hahaha jk :P

15. mathmath333

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.

16. ikram002p

you wanna prove this ?

17. mathmath333

yes with example

18. ikram002p

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

19. mathmath333

but C is not there in the question

20. ikram002p

sorry i ment to say x :P

21. ikram002p

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

22. ikram002p

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

23. thomas5267

Is it possible to do something like this? $A\cup X=B\cup X\land A\cap X=B\cap X=\emptyset\implies A\cup X\setminus X=B\cup X\setminus X\implies A=B$