Show that if $A$ and $B$ are sets, then $(A\cap B) \cup (A\cap \overline{B})=A$.

Show that if $A$ and $B$ are sets, then $(A\cap B) \cup (A\cap \overline{B})=A$.

So I have to show that $(A\cap B) \cup (A\cap \overline{B})\subseteq A$ and that $A \subseteq(A\cap B) \cup (A\cap \overline{B})$.

Lets begin with the first one:

If $x \in (A \cap B)$ it means $x \in A \wedge x \in B$.

If $x \in (A \cap \overline{B})$ it means $x \in A \wedge x \in \overline{B}$.

And the second one:

If $x \in A$ it means $x \in (A \cap B)$.

But here after I am confused.

Hint: $(A\cap B)\cup(A\cap\overline{B})=A\cap(B\cup\overline{B})=A$.

Following your approach of chasing elements, you can say If $x \in (A \cap \overline{B})$ it means $x \in A \wedge x \in \overline{B}$, so $x \in A \wedge x \not \in {B}$. Therefore $x \in ((A\cap B) \cup (A\cap \overline{B}))$ means $(x \in A \wedge x \in {B})\vee (x \in A \wedge x \not \in {B})$ and use the distributive principle