**a logical equivalence is derived from Theorem**

**2.1.1. Supply a reason for each step.**

**48. (p ∧ ∼q) ∨ (p ∧ q) ≡ p ∧ (∼q ∨ q) by (a)**

**≡ p ∧ (q ∨ ∼q) by (b)**

**≡ p ∧ t by ©**

**≡ p by (d)**

**Therefore, (p ∧ ∼q) ∨ (p ∧ q) ≡ p.**

**Answer:**

**a logical equivalence is derived from Theorem**

**2.1.1. Supply a reason for each step.**

**48. (p ∧ ∼q) ∨ (p ∧ q) ≡ p ∧ (∼q ∨ q) by (a)**

**≡ p ∧ (q ∨ ∼q) by (b)**

**≡ p ∧ t by ©**

**≡ p by (d)**

**Therefore, (p ∧ ∼q) ∨ (p ∧ q) ≡ p.**

**Answer:**