Proving Set Equivalencies

Question

I am currently trying to prove the following proposition: $A \equiv (A - B) \cup (A \cap B)$

So far, I have the following proof:

$x \in (A - B) \cup (A \cap B) \\ \implies x \in (A - B) \lor x \in (A \cap B) \\ \implies x \in (A \cap B^\complement) \vee x \in (A \cap B) \\ \implies x \in A \\ \implies (A - B) \cup (A \cap B) \subseteq A$

Assuming I have the first part right, how would I go about proving that $A \subseteq (A - B) \cup (A \cap B)$? Thank you!

Answer 1

A try:

Let $\Omega$ be the Universe (i. e. $A, B, \overline{B} \subseteq \Omega$).

$x \in A \\ \implies x \in A \land x \in \Omega \\ \implies x \in A \land (x \in \overline{B} \lor x \in B) \\ \implies (x \in A \land x \in \overline{B}) \lor (x \in A \land x \in B) \\ \implies x \in A \cap \overline{B} \lor x \in A \cap B \\ \implies x \in A - B \lor x \in A \cap B \\ \implies x \in (A - B) \cup (A \cap B) $

Like in your proof, I used the distributivity of $\lor$.