( ) A statement is said to be a contradiction if its truth value is always F irrespective of the truth values of its component statements. It is denoted by F. ( ) A statement which is neither a tautology nor a contradiction is called contingency. ( ) Any two compound statements A and B are said to be logically equivalent or simply equivalent if the columns corresponding to A and B in the truth table have identical truth values. The logical equivalence of the statements A and B is denoted by A ≡ B or A ↔ B . Further note that if A and B are logically equivalent, then A ↔ B must be a tautology. ( ) Some laws of equivalence: Idempotent Laws: (i) p ∨ p ≡ p (ii) p ∧ p ≡ p . Commutative Laws: (i) p ∨ q ≡ q ∨ p (ii) p ∧ q ≡ q ∧ p . Associative Laws: (i) p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r (ii) p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r . Distributive Laws: (i) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) (ii) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) Identity Laws: (i) p ∨ T ≡ T and p ∨ F ≡ p (ii) p ∧ T ≡ p and p ∧ F ≡ F Complement Laws : (i) p ∨ ¬p ≡ T and p ∧ ¬p ≡ F (ii) ¬ T ≡ F and ¬ F ≡ T Involution Law or Double Negation Law: ¬(¬p ) ≡ p De Morgan’s Laws: (i) ¬(p ∧ q ) = ¬p∨¬q (ii) ¬(p∨q ) = ¬p∧¬q Absorption Laws: (i) p∨(p∧q )≡p (ii) p∧(p∨q )≡p
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Discrete Mathematics