truth tables, logical equivalence, biconditional

Liberal Arts Mathematics I

Class Notes, 2/01/99

The difference between the statement -p ^ -q and the statement -(p ^ q) is seen by comparing the two statements using a truth table.

The truth table depicted below shows that the statements have different truth values in the second and third rows, when p and q have different truth values.

Since the columns differ the statements are not equivalent.

The figure below depicts the equivalence of the statements -p ^ -q and -(p v q), using the <=> symbol for equivalence.

The figure also depicts the non-equivalence of the statements tested in the above truth table.

Note the similarity with de Morgan's Laws.

The truth table depicted below evaluates the given expression.

We first evaluate the -q, then the ^, then the v symbols.

Consider the statement in the figure below. We wish to evaluate the truth of this statement for different truth values of 'work hard' (W) and 'pass' (P).

We consider the statement true unless we are certain that it is false.
The only situation in which we are certain that the statement is false is that in which you indeed work hard but fail to pass.
This reasoning is reflected in the truth table.

The conditional p -> q, read 'if p then q', is false only when the p statement is true and the q statement is false. In any other situation the statement is true, as indicated in the truth table below.

The biconditional p <-> q means (p -> q) and (q -> p), as indicated below.

This statement is often read 'p if and only if q'.
The truth values for the statement are easily evaluated by a truth table, with the indicated result.

The statements A and B in the figure below are not logically equivalent, since if you are not nice but still get to eat it follows that statement A is true while statement B is false.

We can thus see that 'if p then q' has the same meaning as 'p only if q', but is not logically equivalent to 'q only if p'.