Use the properties to verify the logical equivalences in the following. Supply a reason for each step. a. (p ∧∼ q) ∨ p ≡ p b. p ∧ (∼ q ∨ p) ≡ p c. ∼ (p ∨∼ q) ∨ (∼ p ∧∼ q) ≡ ∼ p d. ∼ ((∼ p ∧ q) ∨ (∼ p ∧∼ q)) ∨ (p ∧ q) ≡ p e. (p ∧ (∼ (∼ p ∨ q))) ∨ (p ∧ q) ≡ p

a. $(p \land \sim q) \lor p \equiv$ [By the Absorption law] $\equiv (p \land \sim q) \lor (p \lor (p \land q))\equiv$ [By the Associative and Commutative laws] $\equiv ((p \land \sim q) \lor (p \land q)) \lor p ) \equiv$ [By the Distributive law] $\equiv ((p \land (\sim q \lor q)) \lor p)$ $\equiv$ [By the Negation law] $\equiv ((p \land t) \lor p)$ $\equiv$ [By the Identity law] $\equiv p \lor p \equiv$ [By the Idempotent law] $\equiv p$

b. $p \land (\sim q \lor p) \equiv$ [By the Distributive law] $\equiv (p \land \sim q)\lor (p \land p) \equiv$ [By the Idempotent law] $\equiv (p \land \sim q)\lor p \equiv$ [Equivalent to expression a.] $\equiv p$

c. $\sim (p \lor \sim q) \lor (\sim p \land \sim q) \equiv$ [By the De Morgan's law] $\equiv \sim (p \lor \sim q) \lor \sim (p \lor q) \equiv$ [By the De Morgan's law] $\equiv \sim ((p \lor \sim q) \land \ (p \lor q)) \equiv$ [By the Distributive law] $\equiv \sim (p \lor (\sim q \land q)) \equiv$ [By the Negation law] $\equiv \sim (p \lor c) \equiv$ [By the Identity law] $\equiv \sim p$

d. $\sim ((\sim p \land q) \lor (\sim p \land \sim q)) \lor (p \land q) \equiv$ [By the Distributive law] $\equiv$ $\sim (\sim p \land (q \lor \sim q)) \lor (p \land q) \equiv$ [By the Negation law] $\equiv \sim (\sim p \land t) \lor (p \land q) \equiv$ [By the Identity law] $\equiv p \lor (p \land q) \equiv$ [By the Absorption law] $\equiv p$

e. $( p \land (\sim (\sim p \lor q))) \lor (p \land q) \equiv$ [By the De Morgan's law] $\equiv ( p \land ( p \land \sim q)) \lor (p \land q) \equiv$ [By the Associative law] $\equiv$ $(( p \land p) \land \sim q) \lor (p \land q) \equiv$ [By the Idempotent law] $\equiv ( p \land \sim q) \lor (p \land q) \equiv$ [By the Distributive law] $\equiv p \land (\sim q \lor q) \equiv$ [By the Negation law] $\equiv p \land t$ $\equiv$ [By the Identity law] $\equiv p$