3.4 QUANTIFICATION OF PROPOSITIONAL FUNCTIONS IN SEVERAL VARIABLES 161
[Hints: In (a) and (d), look at the logical negation of both sides, recalling that p is
stronger than q if and only if -q is stronger than -p[by Theorem l(n), Article
2.31. In (b) and (c), consider what each statement means with respect to the propo-
sition r and the truth set Q of q(x).]
- Recall from Theorem l(t), Article 2.3, that [(p A q) -* r] - [(p -, r) v (q -+ r)] is
a tautology. Let U equal the set of all real valued functions of a real variable. Let
propositional functions in the variable f, p( f ), q( f ), and r( f) be defined by:
pw): f has a relative maximum at x = 0
qu): f is differentiable at x = 0
With these substitutions, the left side of the preceding tautology seems to say "func-
tions that have a relative maximum at x = 0 and are differentiable at x = 0 have
derivative zero at x = 0" (which is Ee). The right side would appear to represent
"either functions that have a relative maximum at 0 have derivative zero at x = 0
or functions that are differentiable at 0 have derivative zero at x = 0." Both of these
are false so that their disjunction is false. Resolve this apparent contradiction.
3.4 Quantification of Propositional
Functions in Several Variables
Expressions such as "d, is perpendicular to d," (U = the set of all lines in
a given plane), "x I y" or "x + y = z" (U = R), and ''f = g 0 h" (U = the set
of all real-valued functions of a real variable), are examples of propositional
functions in two or more variables. If we denote "x + y = z" by the symbol
p(x, y, z), then p(5, - 3,2) is a true statement and p(4,0, 5) is false. On the
other hand, p(5, y, z) and (3x)(p(x, y, z)) are not statements, but rather, they
are propositional functions in two variables, whereas p(5, y, 7) is a proposi-
tional function in the single variable y. Each of the last three expressions,
of course, is neither true nor false as it stands.
Assuming that a set Ui constitutes the domain of discourse for the ith
of the n variables of a propositional function p(x,, x2,... , xn), the Cartesian
product U, x U, x - -. x Un is the domain of discourse for p. In particular,
if U is a common domain for all n variables, then Un = U x U x... x U
(n times) is the domain of p, so that the truth set of any such predicate is
a subset of Un.
There is a greater variety of ways to make a propositional function into
a statement, for functions of several variables, than was the case for functions
of a single variable. We may, of course, substitute for each of the variables.
But then again, we might substitute for all variables except one and mod-
ify the remaining "one variable predicate" with a quantifier. Letting p(x, y, z)
stand for the equation x + y = z and letting r(y) = p(5, y, 7), we find that
the quantified predicate (3y)(r(y)) or (3y)(p(5, y, 7)) is the (true) assertion that
the equation 5 + y = 7 has a real solution. Here is another example of this
situation.
