600 Appendix A • Logic and Proofs
( d) Let the domain of x be the set of all real numbers. The given statement
is
,...., 3 x 3 x^2 + x + 1 = 0.
The statement is true for this domain, since the discriminant of this quadratic
expression is negative. If we change the domain of x to be the set of com-
plex numbers, the statement would change its truth-value to false, since the
-1 ± i\/'3
quadratic formula tells us that x^2 + x + 1 = 0 when x =
2
. D
QUANTIFIER NEGATIONQuantified statements occur so frequently in mathematics that it is im-
portant to be able to form their negations correctly. The procedure may seem
tricky at first, but the idea is very simple, and with a little practice you will
become quite good at it. The following principle is extremely important.
Principle of Quantifier Negation:
(1),...., (Vx, P(x)) = 3 x 3 ,...., P(x).
(2),...., (3x 3 P(x)) = Vx, ,...., P(x).To form the negation of a quantified statement, we simply change the quantifier
(universal to existential, or vice versa) and negate the statement that follows
the quantifier. A few examples will help make this principle clear.
Examples A.2.12 Negations of quantified statements:
(a),...., Vx, [P(x)!\ Q(x)] = 3x 3,...., [P(x)!\ Q(x)]
= 3 x 3 [,...., P(x)V,...., Q(x)]
(b),...., 3 x, [P(x) =} Q(x)] = Vx,,...., [P(x) =} Q(x)]
= Vx, [P(x)!\,...., Q(x)]
(quantifier negation)
(de Morgan's law)
(quantifier negation)
(negation of=?, A.l.22) DExamples A.2.13 Form the negation of each given statement, first in sym-
bolic form, then in English where appropriate (see Examples A.2.5-A.2.11):
(a) Vx, sin^2 x + cos^2 x = l.
(b) Vx, I sin xi :S: l.
( c) Every analysis student is intelligent and good-looking.
(d) The square of every nonzero real number is positive.
( e) 3 x 3 sin x = l.