A.2 The Logic of Predicates and Quantifiers 601(f) Somebody stole my wallet.
(g) The equation x^2 - x - 6 = 0 has a solution in the real number system.Solution: In each case we apply the principle of quantifier negation.
(a) The negation is 3 x 3 ,.._, [sin^2 x + cos^2 x = 1], which is equivalent to
3x 3 sin^2 x + cos^2 x-/= 1.
(b) The negation is 3 x 3 "" I sin x i ::::; 1, which is equivalent to 3 x 3
I sin xi > 1.
(c) From Example A.2.6 (b), we want,.._, [Vx, {A(x) =} [J(x) /\ G(x)]}J. This
translates directly into English as, "It is not true that every mathematics stu-
dent is intelligent and good-looking." Using the principle of quantifier negation
and other rules of logic, we can continue:
3x 3 ,.._, {A(x) =} [J(x) /\ G(x)]} (quantifier negation)
3x 3 {A(x) /\ ,.._, [I(x) /\ G(x)]} (negation of implication-A.1.22)
3x 3 {A(x) /\ [,.._, I(x) V ,.._, G(x)]} (de Morgan's law-A.1.18)In English, the resulting negation is, "There is an analysis student who is either
not intelligent or not good-looking." (This is false, of course!)
(d) From Example A.2.6 (c), we want "" [\Ix, [x-/= 0 =} x^2 >OJ], which is
equivalent to
3 x 3,.._, [x -/= 0 =} x^2 > OJ
3 x 3 [x-/= 0 /\ "" (x^2 > O)J
3 x 3 [x -/= 0 /\ x^2 ::::; OJ(quantifier negation)
(negation of implication-A.1.22)
(property of real numbers)In English, "There is a nonzero real number whose square is not positive."
(e) The negation,,.., (3x 3 sinx = 1) , is equivalent to \Ix, sinx-/= 1.
(f) From Example A.2.11 (a), we want,.., [3x 3 S(x)], which is equivalent
to \Ix,,.._, S(x). In smooth English we say simply, "Nobody stole my wallet."
(g) From Example A.2.11 (c), we want
,.., [3x 3 x^2 - x - 6 = o] =\Ix,,.., (x^2 - x - 6 = o) (quantifier negation)
= \Ix, x^2 - x - 6 -/= 0.
In smooth English, it is best to say simply, "There is no real number x that
satisfies the equation x^2 - x - 6 = O." 0
CATEGORICAL PROPOSITIONSBefore the creation of modern symbolic logic around the beginning of the
last century, classical logic had identified four "categorical propositions." Not