1549901369-Elements_of_Real_Analysis__Denlinger_

A.2 The Logic of Predicat es and Quantifiers 597

symbolized P(x, y), Q( u, v), etc. Of course, we can have a propositional function

in any number of variables. For example, we could have:

P(x) = 7x - 24 = sin3x

Q(x,y) = 5x-ll > 8y

3x-4y

R(x,y,z) = 2 = 17

x + 15 z

Using Logical Connectives: Even though propositional functions are
not propositions, we will allow them to be j oined together into compound ex-
pressions using the five logical connectives discussed in Section A.l. Thus, we
will allow such expressions as

P(x) =? Q(x)
P(x) f\ [Q(x, y) V ,._, R(y, z)].
Of course, these are not propositions, since they do not have a truth-value as
long as the variables are unknown. They become propositions whenever the
variables are replaced by constants from their respective domains. They can
also be made into propositions by a process called "quantification,'' to which
we now turn our attention.

UNIVERSAL QUANTIFICATION

Definition A.2.4 (Universal Quantification) Suppose a propositional func-
tion P(x) is true for all values of x in its domain. That fact is itself a proposition
that we denote

\:Ix, P(x).

This is read "for all x , P(x)." The symbol "\:Ix" is called the universal
quantifier. Note again that "\:Ix, P (x)" is a proposition. It is true or false,
even though P(x) is not by itself a proposition.

Examples A.2.5 Some universally quantified stat ements. (Assume the do-
main is the set of all real numbers.)

(a) \:Ix, sin^2 x + cos^2 x = 1

(b) \:Ix, I sin xi :S 1

(c) \:Ix, 6x+ 11=7

(true);

(true);

(false). D