Sort on true or false this sentence:

(A-B)∪(A∩B) = A

Here's another proof. To prove two sets equal we need to prove two things:

1) The left side is a subset of the right side; and

2) The right side is a subset of the left side.

Let's do (2) first. Suppose $a \in A$. Then $a \in A - B$; and since $a$ is an element of one of the disjuncts of a union, it's in the union. Therefore the right side is a subset of the left side.

Now (1). Say $x \in (A - B ) \cup (A \cap B)$. Either $x \in A - B$ or $x \in A \cap B$.

If $x \in A - B$ then $a \in A$ by definition of set difference and we're done.

If $x \in A \cap B$ then $x \in A$ by def of intersection. Either way $x \in A$ and we're done.