EXAMPLE 2.2.6

Complete the following truth table.

SOLUTION

We fill in column 5 by comparing column 3 with column 2 and using conjunction. The only time this "and" statement is true is when columns 3 and 2 are both true.

We fill in column 6 by comparing column 1 with column 4 and using the fact that column 1 is the antecedent for this conditional statement. The only time column 6 can be false is when column 1 is true while column 4 is false.

We fill in column 7 by comparing column 5 with column 2 and using the fact that column 5 is the antecedent for this conditional statement. The only time column 7 can be false is when column 5 is true while column 2 is false.

We fill in column 8 by comparing column 6 with column 5 and using disjunction. The only time this "or" statement can be false is when columns 6 and and 5 are both false.

Note that the truth table truth table shows that the statement in column 7 is a tautology, while the statement in column 6 is equivalent to the statement in column 8.