EXAMPLE 2.1.15

Complete the following truth table.

We will label each of the columns.

To fill in column 5, we compare column 1 with column 4, using the conjunction:

To fill in column 6, we compare column 3 with column 2, using the disjunction:

To fill in column 7, we reverse all of the entries from column 6:

To fill in column 8, we compare column 3 with column 5, using disjunction:

To fill in column 9, we compare column 2 with column 7, using conjunction:

Now that the truth table is complete, we see that it shows one logical equivalency: the statement in column 5 is equivalent to the statement in column 7.

The truth table does not show any tautologies. However, the statement in the last column is always false; such a statement is sometimes called a contradictory.

Identify any equivalencies or tautologies.