Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variable. The logical equivalence of statement forms P and Q is denoted by writing PÛQ.

Two statements are called logically equivalent if, and only if, when the same statement variables are used to represent identical component statements, their forms are logically equivalent.

To test whether two statement forms P and Q are logically equivalent:

1. Construct the truth table for P.
2. Construct the truth table for Q using the same statement variables for identical component statements.
3. Check each combination of truth values of the statement vaiables to see whether the truth value of P is the same as
   the truth value of Q.
   1. If in each row the truth values for P are the same as the truth values for Q, then they are logically equivalent.
   2. If in some row P has a different truth value from Q, then they are not logically equivalent.

Alternatively, a single counter example may be used to show that the statement forms are not logically equivalent.

Ex. 1

Construct a truth table to show that p is logically equivalent to p Ù (p Ú q).

p	q	pÙ ( p Ú q)
T	T	T T  T
T	F	T T  T
F	T	F F  T
F	F	F F  F

Now the underlined column is identical to the p column of truth values so p Ù (p Ú q) is logically equivalent to p.