We are required to prove the following identities using properties of sets:
(i) \[ A \cup (A \cap B) = A \] (ii) \[ A \cap (A \cup B) = A. \]
Proof of (i):
Starting with the left-hand side,
\[ A \cup (A \cap B). \]
Using the distributive law of union over intersection,
\[ A \cup (A \cap B) = (A \cup A) \cap (A \cup B). \]
Using the idempotent law,
\[ A \cup A = A. \]
Therefore,
\[ A \cup (A \cap B) = A \cap (A \cup B). \]
Using the absorption law,
\[ A \cap (A \cup B) = A. \]
Hence,
\[ A \cup (A \cap B) = A. \]
Proof of (ii):
Starting with the left-hand side,
\[ A \cap (A \cup B). \]
Using the distributive law of intersection over union,
\[ A \cap (A \cup B) = (A \cap A) \cup (A \cap B). \]
Using the idempotent law,
\[ A \cap A = A. \]
Therefore,
\[ A \cap (A \cup B) = A \cup (A \cap B). \]
Using the absorption law,
\[ A \cup (A \cap B) = A. \]
Hence,
\[ A \cap (A \cup B) = A. \]
Thus, both the identities are proved.