Prove that for any sets $A$ and $B$,$A \cup (A \cap B) = A$.
(A) To prove: $A \cup (A \cap B) = A$
We know that for any sets $A$ and $B$,the intersection $A \cap B$ is a subset of $A$,i.e.,$(A \cap B) \subset A$.
Also,$A \subset A$.
Therefore,the union of these two sets is $A \cup (A \cap B) \subset A$ ........... $(1)$
Conversely,any element $x \in A$ implies $x \in A \cup (A \cap B)$ by the definition of union,so $A \subset A \cup (A \cap B)$ ........... $(2)$
From $(1)$ and $(2)$,by the definition of set equality,we conclude that $A \cup (A \cap B) = A$.
We know that for any sets $A$ and $B$,the intersection $A \cap B$ is a subset of $A$,i.e.,$(A \cap B) \subset A$.
Also,$A \subset A$.
Therefore,the union of these two sets is $A \cup (A \cap B) \subset A$ ........... $(1)$
Conversely,any element $x \in A$ implies $x \in A \cup (A \cap B)$ by the definition of union,so $A \subset A \cup (A \cap B)$ ........... $(2)$
From $(1)$ and $(2)$,by the definition of set equality,we conclude that $A \cup (A \cap B) = A$.
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