596 Appendix A • Logic and Proofs
Such statements are sometimes called "open sentences" -the variables re-
serve open spaces in the sentence into which we may later put sp ecific constants.
Logicians prefer to call such statements "predicates." For example, in the state-
ment "x > 3," we can think of x as the subject, and " > 3" as the predicate.
Examples A.2.2 Some propositional functions (open sentences):
(a) 5x^2 + 2x - 7 = 0.
(b) sin4x = 0.5.
(c) n is divisible by 5.
(d) 3x - 4y = 0.
(e) x^2 - 4 = (x - 2)(x + 2).
sinx
(f) tanx = --.
cosx
(g) A~ B. DE ach variable appearing in a propositional function has a domain, the set
of all constants (concrete objects) that may be substituted in place of that
variable. In Example A.2.2, the domain of x would be the set of real numbers,
the domain of n, the set of integers, and the domain of A and B , a collection
of sets. The domain of a variable is allowed to change from one application to
another, depending upon the intention of the user.
A propositional function may be true for some values of its variable(s) and
false for others. The truth set of a propositional function in one variable is the
set of all values in the domain of that variable that make it a true proposition.
Examples A.2.3 In each of the following, assume that the domain of x is the
set of all real numbers.
Propositional Function
x^2 - 3x + 2 = 0
x^2 - 3x + 2 > 0
x^2 + 2x + 1 :::; 0
x^2 - 4 = (x - 2)(x + 2)
sin^2 x + cos^2 x = 1
sinx = 3Truth Set
{1,2}
(-oo, 1) U (2, oo)
[-1,2]
(-oo, oo)
(-oo, oo)
0, the empty set^3 DNotation: A propositional function in one variable will be symbolized
P(x), Q(y), etc. Similarly, a propositional function in two variables will be
- The empty set is discussed in Appendix B.l.