Bridge to Abstract Mathematics: Mathematical Proof and Structures

3.2 QUANTIFICATION 89

Second, notice that the statement (Vx)(p(x)) is true precisely when the
statement ha) is true for every possible substitution of a specific object a
from the domain of discourse U, whereas (3x)(p(x)) is true precisely when
p(a) is true for at least one substitution of an object a from U. Third, note
that in translating a symbolized statement (3x)(p(x)) into English, we must
insert the words "such that" or some equivalent formulation (e.g., "for
which") before the translation of p(x). Finally, we note that definitions per-
taining to situations in which quantified variables are restricted to certain
subsets of the universal set U are presented in Exercise 7, Article 3.3.

EXAMPLE 1 Let U = R. Then:

(a) (3x)(x2 = 4) is true, whereas (Vx)(x2 = 4) is false. This is so be-

cause the truth set of the open sentence p(x): x2 = 4 is P = ( - 2,2)

which is, on the one hand, nonempty, but on the other, fails to

equal U.

(h) (Vx)(x2 2 0) is true, as is (3x)(x2 2 0). (Why?)

(c) (Vx)(x2 = - 5) and (3x)(x2 = - 5) are both false, since the truth set

of the predicate "x2 = - 5" is (ZI and U = R # 0.

(d) Can you produce an open sentence Ax), with U = R, for which

(Vx)(p(x)) is true, while (3x)(p(x)) is false? If not, and if no

such predicate exists, what possible theorem of the predicate cal-

culus is suggested? [See Theorem 2(c), Article 3.3.1 Can you think

of a circumstance, involving a different choice of U, that might

allow (Vx)(p(x)) to be true, while (3x)(p(x)) is false? [See Exercise

8(b), Article 3.3.1

ENGLISH TRANSLATIONS OF STATEMENTS

INVOLVING QUANTIFIERS

There are many possible English translations of quantified predicates. Since

you will on occasion need to write a given English sentence in symbolic

form, it is important to become familiar with these translations, some of

which involve the phrases "for every" and "there exists" only implicitly (we

say that such a statement involves a hidden quant$er). Consider the pred-

icate x2 = 4 of part (a) of Example 1. Read literally, (Vx)(x2 = 4) says "for

every x, x squared equals four." This, however, can also be expressed "for

all real numbers x, x2 = 4," or "every (each) real number x has 4 as its

square." Note that, in this last translation, we do not explicitly say "for

every" and do not use any dummy variable. Similarly, (3x)(x2 = 4), which

we read literally "there exists x such that x2 = 4," can also be expressed

"there exists a real number x for which x2 = 4" or "there exists a number

x whose square is 4," or finally, "some real number has 4 as its square."

The existential quantifier is hidden in the last translation. The main point

of these examples is that you learn to associate the universal quantifier

with the words "every," "each," and "all," and the existential quantifier with