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  Mathematical Logic Logical Equivalence Definition 12.20 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  must be  tautology .   Some Laws of Equivalence   1. Idempotent Laws (i)  p  ∨  p  ≡  p (ii)  p  ∧  p  ≡  p  . Proof In the above truth table for both  p , p  ∨  p  and  p   ∧   p  have the same truth values. Hence  p  ∨  p  ≡  p  and  p  ∧  p  ≡  p  .   2. Commutative Laws (i)  p   ∨   q   ≡   q   ∨   p (ii)  p   ∧   q   ≡   q   ∧   p  . Proof  (i) The columns corresponding to  p   ∨   q  and  q   ∨   p  are identical. Hence  p   ∨   q   ≡   q   ∨   p  . Similarly (ii)  p   ∧   q   ≡   q   ∧   p  can be proved.   3. Associative Laws (i)  p   ∨