To prove: The following properties of logical equivalence. Idempotence: (i) ( p ∨ p ) ⇔ p (ii) ( p ∧ p ) ⇔ p Commutativity: (i) ( p ∨ q ) ⇔ ( q ∨ p ) (ii) ( p ∧ q ) ⇔ ( q ∧ p ) Associativity: (i) ( ( p ∨ q ) ∨ r ) ⇔ ( p ∨ ( q ∨ r ) ) (ii) ( ( p ∧ q ) ∧ r ) ⇔ ( p ∧ ( q ∧ r ) ) Distributivity: (i) ( p ∨ ( q ∧ r ) ) ⇔ ( ( p ∨ q ) ∧ ( p ∨ r ) ) (ii) ( p ∧ ( q ∨ r ) ) ⇔ ( ( p ∧ q ) ∨ ( p ∧ r ) ) Double Negation: ¬ ( ¬ p ) ⇔ p De Morgan’s Laws: (i) ¬ ( p ∨ q ) ⇔ ( ( ¬ p ) ∧ ( ¬ q ) ) (ii) ¬ ( p ∧ q ) ⇔ ( ( ¬ p ) ∨ ( ¬ q ) ) (i) ( p ∨ 1 ) ⇔ 1 (ii) ( p ∧ 1 ) ⇔ p (i) ( p ∨ 0 ) ⇔ p (ii) ( p ∧ 0 ) ⇔ 0 (i) ( p ∨ ( ¬ p ) ) ⇔ 1 (ii) ( p ∧ ( ¬ p ) ) ⇔ 0 (i) ¬ 1 ⇔ 0 (ii) ¬ 0 ⇔ 1 ( p → q ) ⇔ [ ( ¬ q ) → ( ¬ p ) ] ( p → q ) ⇔ [ ( ¬ p ) ∨ q ] ( p ↔ q ) ⇔ [ ( p → q ) ∧ ( q → p ) ]

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Chapter 1.2, Problem 1E

To determine

To prove: The following properties of logical equivalence.

Idempotence:

(i) (p∨p)⇔p

(ii) (p∧p)⇔p

Commutativity:

(i) (p∨q)⇔(q∨p)

(ii) (p∧q)⇔(q∧p)

Associativity:

(i) ((p∨q)∨r)⇔(p∨(q∨r))

(ii) ((p∧q)∧r)⇔(p∧(q∧r))

Distributivity:

(i) (p∨(q∧r))⇔((p∨q)∧(p∨r))

(ii) (p∧(q∨r))⇔((p∧q)∨(p∧r))

Double Negation:

¬(¬p)⇔p

De Morgan’s Laws:

(i) ¬(p∨q)⇔((¬p)∧(¬q))

(ii) ¬(p∧q)⇔((¬p)∨(¬q))

(i) (p∨1)⇔1

(ii) (p∧1)⇔p

(i) (p∨0)⇔p

(ii) (p∧0)⇔0

(i) (p∨(¬p))⇔1

(ii) (p∧(¬p))⇔0

(i) ¬1⇔0

(ii) ¬0⇔1

(p→q)⇔[(¬q)→(¬p)]

(p→q)⇔[(¬p)∨q]

(p↔q)⇔[(p→q)∧(q→p)]

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