1549901369-Elements_of_Real_Analysis__Denlinger_

598 Appendix A • Logic and Proofs

Examples A.2.6 Express each of the following English sentences in symbolic
form:

(a) Everyone must breathe and eat, or die.

(b) Every analysis student is intelligent and good-looking.

(c) The square of every nonzero real number is positive.

Solution:
(a) Let the domain of x be the set of all people, B(x) = x breathes, E(x) =
x eats, and D(x) = x dies. Then the given statement is

Vx, {[B(x) /\ E(x)] V D(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

Vx, {A(x) ==> [I( x) /\ G(x)]}.

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

Definition A.2.7 Restricted Universal Quantification: If Sis a subset
of the domain of a variable x, then the statement "For all x in the set S, P(x)
is true," is symbolized

Vx ES, P(x).

Example A.2.8 If S denotes the set of all analysis students then the state-
ment, "Every analysis student is intelligent and good-looking" could be sym-
bolized

Vx ES, [I(x)/\ G(x)].

Compare this with Example A.2.6(b) above. D

EXISTENTIAL QUANTIFICATION

Definition A.2.9 (Existential Quantification) Suppose a propositional func-
tion P(x) is true for some (at least one) value of x in its domain. That fact is
itself a proposition, which we denote

3x 3 P(x).