1549901369-Elements_of_Real_Analysis__Denlinger_

A.2 The Logic of Predicates and Quantifiers 599

This is read "there exists an x such that P( x)." The symbol ":3 x" is called

the existential quantifier, and the (optional) symbol 3 is present to indicate

the phrase "such that." Note that ":lx 3 P(x)" is a proposition; it is true or

false, even though P(x) is not by itself a proposition.

We can also use restricted existential quantification to symbolize

"there exists an x in the set S such that P(x) is true" as

:Jx ES 3 P(x).

Examples A.2.10 Some existentially quantified statements:

(a) :Jx 3 sinx = 1

(b) :3 x 3 sin x 2: 2

( c) :3 x 3 6x + 11 = 7

(true);

(false);

(true). D

Examples A.2.11 Express each of the following English sentences in symbolic
form:

(a) Somebody stole my wallet.
(b) Some analysis students are intelligent and good-looking.
(c) The equation x^2 - x - 6 = 0 has a solution in the real number system.
(d) The equation x^2 + x + 1 = 0 has no real number solution.
Solution:
(a) Let the domain of x be the set of all people, and S(x) = x stole my
wallet. Then the given statement is

:3 x 3 x stole my wallet, or

:Jx 3 S(x).

(b) Let the domain of x be the set of all people, A(x) = x is an analysis
student, I(x) = x is intelligent, and G(x) = x is good-looking. The given
statement is

:Jx 3 {A(x) /\ [I(x) /\ G(x)]}.

Alternatively, we could let S = the set of all analysis students; then the
statement becomes

:Jx ES 3 [J(x)/\ G(x)].

(c) Let the domain of x be the set of all real numbers. The given statement
is

:3 x 3 x^2 - x - 6 = 0.

The statement is true, since x^2 - x - 6 = 0 is true when x = -2 or x = 3.