. . Logical Equivalence Definition . 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 ⇔ From the definition, it is clear that, if A and B are logically equivalent, then A B Û must be a tautology . Some Laws of Equivalence . Idempotent Laws (i) p ≡ (ii) p ≡ Proof ∨∨ ∧∧ Table . In the above truth table for both p , p and p have the same truth values.
Hence ≡ and p ≡ . Commutative Laws (i) p ≡ (ii) p ≡ Proof (i) ∨∨ ∨∨ Table . The columns corresponding to p and q are identical. Hence p ≡ Similarly (ii) p ≡ can be proved.
. Associative Laws (i) p ) ≡ ) ∨ (ii) p ) ≡ ) ∧ . - - Discrete Mathematics Proof The truth table required for proving the associative law is given below. ∨∨ ∨∨ ∨∨ (( )) ∨∨ ∨∨ ∨∨ (( )) Table .
The columns corresponding to p ) ∨ and p ) are identical. Hence p ) ≡ ) ∨ . Similarly, (ii) p ) ≡ ) ∧ can be proved. .
Distributive Laws (i) p ≡ (ii) p ≡ Proof (i) ∧∧ ∨∨ ∧∧ ∨∨ ∨∨ ∨∨ ∧∧ ∨∨ Table . The columns corresponding to p ) and ( are identical. Hence ≡ ) . Similarly (ii) p ≡ ) can be proved.
. Identity Laws