1. **State the problem:** Prove the identities using algebraic laws of sets: (i) $ (A \cup B) \cap (A \cup B^c) = A $ (ii) $ A \cup (A \cap B) = A $ --- 2. **Recall important laws:** - Distributive law: $X \cap (Y \cup Z) = (X \cap Y) \cup (X \cap Z)$ and vice versa. - Complement law: $B \cup B^c = U$ (universal set). - Idempotent law: $A \cup A = A$ and $A \cap A = A$. - Absorption law: $A \cup (A \cap B) = A$. --- 3. **Proof of (i):** Start with the left side: $$ (A \cup B) \cap (A \cup B^c) $$ Use distributive law: $$ = A \cup (B \cap A) \cup (B \cap B^c) $$ Note $B \cap A = A \cap B$ and $B \cap B^c = \emptyset$ (empty set), so: $$ = A \cup (A \cap B) \cup \emptyset = A \cup (A \cap B) $$ By absorption law: $$ = A $$ --- 4. **Proof of (ii):** By absorption law directly: $$ A \cup (A \cap B) = A $$ --- **Final answers:** (i) $ (A \cup B) \cap (A \cup B^c) = A $ (ii) $ A \cup (A \cap B) = A $