Bridge to Abstract Mathematics: Mathematical Proof and Structures

2.2 TAUTOLOGY, EQUIVALENCE, THE CONDITIONAL, AND BICONDITIONAL 61

A consequence of Example 2 is that, if one person claims

and if another claims

It is not the case that either {-1,O) n (0, 1) = (0) or

(- 17 0) n (0, 1) = %

they are both saying the same thing; thus they are either both right or both
wrong. The first statement has the form - p A - q, whereas the second is
structured -(p v q) (What is p? What is q?), two statement forms that we
have seen in Example 2 to be logically equivalent or the same in a logical
sense. Note that both are wrong in their claims since p and q are both true
(row 1 of the table in Figure 2.4).
The main role in mathematics of logical equivalence lies in the idea,
implied in the previous paragraph, that two logically equivalent statement
forms can be thought of as "the same," from the point of view of logic, and
so are interchangeable. Thus if we must prove a statement whose form is
p, and find it easier to prove q, where q is logically equivalent to p, then
we may prove p by proving q. We will see numerous applications of this
principle later in the text.
Can you give other examples of logically equivalent statement forms
using only three connectives -, v , and A? Think about this question;
then see Exercise 3, Article 2.1, and Theorem 1, Article 2.3.

THE CONDITIONAL AND BICONDITIONAL

Very few statements with significant mathematical content that are easily
understandable can be formulated by using the connectives and, or, and
not alone (see, however, Exercises 10 and 11, Article 2.3). As noted earlier
in this chapter, most theorems have the form "if... then" or "if and only
if," while every definition, by nature, admits an "if and only if" formulation.
Thus we are led to Definition 3.

DEFINITION 3

Given statements p and 9, we define:

(a) The statement p implies q, denoted p -+ 9, also read "if p, then 9," is

true except in the case where p is true and 9 is false. Such a statement

is called a conditional; the component statements p and 9 are called the

premise and conclusion, respectively.

(b) The statement p if and only if q, denoted p t, 9, also written "p iff 9,"

is true precisely in the cases where p and q are both true or p and q are

both false. Such

The truth tables for

a statement is called a biconditional.

these two connectives are given in Figure 2.5.