Biconditionals and Combinations of Connectives

A sentence of the type

is called a biconditional  , denoted

When P and Q are sentences, the truth table for is:

P Q

T T T

T F F

F T F

F F T

In mathematics the biconditional is encountered in many forms. The following have the same meaning:

• 
• P is equivalent to Q
• P if and only if Q
• Q if and only if P
• P iff Q
• If P, then Q and conversely
• If Q, then P and conversely
• P is a necessary and sufficient condition for Q
• Q is a necessary and sufficient condition for P.

Combinations of , , , , and often occur. A facility at recognizing them is essential for mathematical reading and proof. Consider the following statement:

If p is prime, then if p is even p must be smaller than 7.

This breaks up into three statements:

• P: p is prime.
• Q: p is even.
• R: p must be smaller than 7.

We can then translate the original statement into

If k is perpendicular to and is perpendicular to m, then k is parallel to m.

Let

• P: k is perpendicular to
• Q: is perpendicular to m
• R: k is parallel to m.

Then the sentence translates as