Use truth tables to verify that each of the following is a tautology. Parts (a) and (b) are called commutative laws, parts (c) and (d) are associative laws, and parts (e) and (f) are distributive laws.
(a) (p ∧ q) ⇔(q ∧ p)
(b) (p ∨ q) ⇔ (q ∨ p)
(c) [p ∧ (q ∧r)] ⇔ [(p ∧q) ∧r]
(d) [p ∨ (q ∨ r)] ⇔ [(p ∨ q) ∨ r]
(e) [p ∧ (q ∨ r)] ⇔ [(p ∧q) ∨ (p ∧r)]
(f) [p ∨ (q ∧r)] ⇔ [(p ∨ q) ∧ (p ∨ r)]
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