Answer
See step by step answer for proof
Work Step by Step
First part- Let $ x \in A$ $\cup$ ($A$ $\cap$ $B$)
Using the definition of union, an element of $A$ $\cup$ ($A$ $\cap$ $B$) is an element that is in $A$ or in $A$ $\cap$ $B$.
$$ (x \in A) \lor (x \in (A \cap B)) $$
Using the definition of intersection, an element of $A \cap B$ is an element that is in A and B. Thus if $x \in A \cap B$, then $x \in A$.$$x \in A \lor x \in A$$By the idempotent law:$$x \in A$$
Thus, we have shown that $A$ $\cup$ ($A$ $\cap$ $B$) $\subseteq $ $A$
Conversely,
Let $ x \in A$, then according to definition of union $x \in $ $A$ $\cup$ ($A$ $\cap$ $B$).
We have shown that $ A \subseteq $ $A$ $\cup$ ($A$ $\cap$ $B$).
We have obtained that $ A \subseteq $ $A$ $\cup$ ($A$ $\cap$ $B$) and $A$ $\cup$ ($A$ $\cap$ $B$) $\subseteq A$.
Hence $A$ $\cup$ ($A$ $\cap$ $B$) $=A$
