588 Appendix A • Logic and Proofs
Thus, P <=> Q is true only when P and Q have the same truth value.
Sometimes we say, "Pis necessary and sufficient for Q." Mathematicians have
invented the short word "iff"^2 to stand for "if and only if." That is,
"P iff Q" means "P <=> Q ."We shall use "iff" frequently in this course.Examples A.1.10 Some true bi-implications:
(a) x^2 = 4 <=> x = 2 or x = -2.
(b) Ix - 31 < 1 iff 2 < x < 4.
(c) A triangle is isosceles if and only if two of its angles are congruent. DCAUTION: In ordinary conversation, people usually avoid the "if and only
if" phrase, because it is awkward. They often say "if" when they really mean "if
and only if." For example, a person who says, "If you mow my yard, I'll give
you $25" probably means "I'll give your $25 if and only if you mow my yard."
(The $25 will not likely be paid if the yard is not mowed!)
In mathematics we must be very careful not to write "if" when we mean
"if and only if. " That is perhaps a good reason for using the word "iff."
Remark A.1.11 (Use of "if" in Definitions) When stating definitions,
mathematicians usually use "if" to mean "if and only if. " It is logically incor-
rect usage of "if" but sanctioned by long-standing practice. According to this
practice, definitions such as the following are quite common:
Definition: A triangle is isosceles if it has (at least) two con-
gruent sides.While it would seem to be bad style to use "if" instead of the more correct
"iff" in definitions, mathematicians seem to be incurable of this habit.
Definition A.1.12 The negation of a proposition Pis the proposition "not-
P." It is symbolized ,.._, P and is defined by the truth-table:
- One bit of mathematical folklore suggests that the word "iff" was coined in the 1950s when
mathematicians first conjoined "it'' with "fi" (backwards "it'') to form the biconditional "iffi,"
and then dropped the final "i" in the interest of simplicity. ·