Prove $(A \cup B) \cap (B \cup C) \cap (C \cup A) = (A \cap B) \cup (A \cap C) \cup (B \cap C) $

Question

Prove the set relation without using Venn diagrams:
$$(A \cup B) \cap (B \cup C) \cap (C \cup A) = (A \cap B) \cup (A \cap C) \cup (B \cap C) $$ I have proven that the RHS leads to LHS, but not the other way round. Help!

Answer 1

another approach is to use set theory identities , use the identity : $X\cap(Y\cup Z)=(X\cap Y)\cup (X\cap Z)$ and : $X\cup(Y\cap Z)=(X\cup Z)\cap(X\cup Y)$ and observe that : $A\cap B\cap C\subseteq (A\cap B)\cup(A\cap C)$ and $A\cap B\cap C\subseteq (B\cap C)\cup(A\cap C)$ thats all what you need to complete your answer ...

Answer 2

Assume $x\in LHS$, then show that $x\in RHS$. Similarly assume $x\in RHS\Rightarrow x\in LHS$.

For example $x\in LHS\Rightarrow x\in$ all of $A\cup B$, $B\cup C$, $C\cup A$. This implies $x$ must belong to at least two of the sets $A,B,C$, Otherwise, suppose $x\not\in B,C$, then $x\not\in B\cup C$. Contradiction.

Now if $x$ belongs to 2 of the sets then obviously it belongs to their intersection. Therefore $x\in RHS$.