Prove that for any sets $A$ and $B$,$A \cap (A \cup B) = A$.
(N/A) To prove: $A \cap (A \cup B) = A$
Using the distributive law of intersection over union:
$A \cap (A \cup B) = (A \cap A) \cup (A \cap B)$
Since $A \cap A = A$ (idempotent law):
$= A \cup (A \cap B)$
Since $(A \cap B) \subseteq A$,the union of $A$ and a subset of $A$ is $A$ itself (absorption law):
$= A$
Using the distributive law of intersection over union:
$A \cap (A \cup B) = (A \cap A) \cup (A \cap B)$
Since $A \cap A = A$ (idempotent law):
$= A \cup (A \cap B)$
Since $(A \cap B) \subseteq A$,the union of $A$ and a subset of $A$ is $A$ itself (absorption law):
$= A$
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