# Prove the following set identity using the laws of set theory

I am not sure how to get started on this one. I know I am supposed to make the second part of the first equation negative, just not sure how.

Let A and B be any sets. Prove the following set identity using the laws of set theory (set identities). Justify each step with the law you used. Missing steps and missing justiﬁcation will be penalized.

$$[A \cup (B\cap C)]\cap([A'\cup(B \cap C)]\cap(B \cap C)') = \emptyset$$

• Let x be in the set. $x \in (B \cap C)'$ so $x \notin (B \cap C)$. But $x \in A U (\B \cap C)$ so $x \in A$. But $x \notin (B \cap C)$ but $x \in A' \cup (B \cap C)$ so $x \in A'$. so $x\in A$ and $x \in A'$. A contradiction. – fleablood Dec 14 '15 at 4:21

## 1 Answer

Suppose there exists $x \in$ the set. Then

$x \in (B \cup C)'$

$x \in A \cup (B \cup C)$ but $x \notin (B \cup C)$ so $x \in A$

$x \in A' \cup (B \cup C)$ but $x \notin (B \cup C)$ so $x \in A'$

So $x \in A \cap A' = \emptyset$.

A contradiction. So the set has no elements. So the set is empty.