Originally Posted by

**Runty** I believe I may have solved this properly, but I have some nagging doubts about my process. Here is the question.

Show whether the set equality $\displaystyle (A-B) \cup (B-A) = (A \cup B) - (A \cap B)$ is true, using set identities.

My process has led me to believe it is false, but I can't shake the feeling that I made a mistake somewhere. Here is my work.

$\displaystyle (A-B) \cup (B-A) = (A \cup B) - (A \cap B)$

$\displaystyle (A \cap \overline{B}) \cup (B \cap \overline{A}) = (A \cup B) \cap (\overline{A \cap B})$

$\displaystyle (A \cap \overline{B}) \cup (B \cap \overline{A}) = (A \cup B) \cap (\overline{A} \cup \overline{B})$ De Morgan's Law

$\displaystyle (A \cap \overline{B}) \cup (\overline{A} \cap B) = (A \cup B) \cap (\overline{A} \cup \overline{B})$ Commutative Law

$\displaystyle (A \cup \overline{A}) \cap (\overline{B} \cup B) = (A \cap \overline{A}) \cup (B \cap \overline{B})$ Distributive Laws (Twice) (this is where I think I screwed up)

$\displaystyle U \cap U = \emptyset \cup \emptyset$ Complement Laws (multiple times)

$\displaystyle U = \emptyset$ Identity Laws

I do not know whether or not my application of the Distributive Laws in that case was correct. Could someone please check this?