A⋃(B⋂C)=(A⋃B)⋂(AUC) prove it?

Question

We can see more easily with Venn diagram. I want to prove it without Venn diagram . I tried like this. A⋃(B⋂C)=(A⋃B)⋂(AUC) . Let x∊A⋃(B⋂C) Then x∊A or x∊B⋂C . First assume that x∊A Then x∊A⋃B and x∊A⋃C .... A few step later. Let x∊(A⋃B)⋂(AUC) so x∊A⋃B and x∊A⋃C.But I cant continue . How can I solve it

Answer 1

You may continue your proof as follows.

If $$x\in (A\cup B)\cap (A\cup C)$$ Then we consider two cases.

Case 1) $x\in A$ which implies $x\in A\cup (B\cap C)$

Case 2) $x\notin A$ then $x$ must be in both $B$ and $C$ therefore it is in $B\cap C$ which makes it $x\in A\cup (B\cap C)$