a) Determine whether are equivalent without using truth table.

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To do so, you need to use the following laws:

Commutative laws: p ∧ q ≡ q ∧ p

p ∨ q ≡ q ∨ p

De Morgan’s laws: ∼(p ∧ q) ≡ ∼p ∨ ∼q

∼(p ∨ q) ≡ ∼p ∧ ∼q

Idempotent laws: p ∧ p ≡ p

p ∨ p ≡ p

Associative laws: (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Distributive laws: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)