3.4 QUANTIFICATION OF PROPOSITIONAL FUNCTIONS IN SEVERAL VARIABLES IOQEXAMPLE 1 Let t(x, y) represent the inequality x S y with domain of
discourse R x R. Describe the truth set of q(x) = t(x, 5). Determine
whether (Vx)(q(x)) and (3x)(q(x)) are true or false.
SOIU tion We can best describe the various truth sets by graphing. The truth
set of t(x, y), call it T, is described in Figure 3.1~. Note that T G R x R.
The truth set of q(x): x I 5 is necessarily a subset of R x (51, since the
value of y has been fixed at 5. In other words, the set R x (51, better
known as the horizontal line with equation y = 5, is the domain of dis-
course for q(x) [see Figure 3.1 b]. The truth set is that portion of R x (5)
that is also in T [see Figure 3. lc], namely, the set ((x, y) 1 x y, y = 5) =
(R x (5)) n T. Since this truth set is nonempty [(3,5) is an element, e.g.1,
then (3x)(q(x)) is true. Since it does not equal all of R x (5) [e.g.,
(7, 5) E R x (51, but is not in T and so is not in the truth set of q(x)], then
(Vx)(q(x)) is false. 0
MIXED QUANTIFIERSAnother possibility for converting p(x, y, z): x + y = z into a proposition is
to quantify all the variables, as in the statements (Vx)(Vy)(Vz)p(x, y, z), a
false statement, and (3x)(3y)(3z)p(x, y, z), which is true. You should have no
difficulty in recognizing the meaning of these statements; the first says that
the given equation is true for any three real numbers, the second states that
it is true at least once, that &Tor some three real numbers.
The meaning may not be so obvious if we write (Vx)(3y)(Vz)p(x, y, z)
or (3x)(3y)(Vz)p(x, y, z); these are examples of mixed quantifiers and are
in general much more difficult to interpret than our first two examples.
One difficulty comes in trying simply to formulate such symbolized state-
ments into meaningful English sentences. Three keys to this are: (1) Insert
"such that" or "having the property that" after any occurrence of 3 that is
followed directly by V or by the predicate. (2) Insert "and" between any two
occurrences of the same quantifier. (3) Whenever (3y) follows (Vx), read
"to every x, there corresponds at least one y" rather than "for every x, there
exists a y." We will explore in detail the reasons for the latter interpretation
after Theorem 1. Thus we read (Vx)(3y)(Vz)p(x, y, z) as "to every x, there
corresponds at least one y such that, for every z, p(x, y, z)." The more com-
plicated expression (3~)(3w)(Vx)(Vy)(3z)pfu, w, x, y, z) is read "there exist v
and w having the property that, to every x and y, there corresponds at least
one z such that p(v, w, x, y, z)."
It is in dealing with mixed quantifiers that the true nature of multiple
quantification, namely, a series of single quantifications, becomes both ap-
parent and important.
EXAMPLE 2 Given a propositional function p(x, y), describe graphically
conditions for the truth of (Vx)(3y)(p(x, y)) and (3y)(Vx)(p(x, y)).