Proving/Disproving set identity $(A\cap B)\cup C= A\cap (B\cup C)$

Question

For any sets $A$, $B$, $C$; $(A\cap B)\cup C= A\cap (B\cup C)$

I understand that this means that (A and B) or C = A and (B or C), but how would you prove or disprove these set identities. Any help would be appreciated, Thanks

Answer 1

Easiest way is normaly using logic expressions:

$$ (A\cap B) \cup C = \{x|\ x\in A,x\in B \lor x\in C \}$$

But in your given example you just have to think about it.

Let's assume $A=\{0\}, B=\{0,1\}, C=\{2\}$ :

$$ (A\cap B)\cup C = \{0,2\} \not = A\cap(B\cup C) = \{0\}$$