3.3 THEOREMS ABOUT PREDICATES IN ONE VARIABLE 95(c) r(x): x2 < 25
*(e) t(x): 3 < x < 5
(g) u(x): x2 I 1
(i) y(x):^0 < 1x1 <^2*(d) s(x): Jx -^41 <^1
(f) u(x): 1x1 5 0
(h) w(x): (x 2 0) A (x 1 0)Theorems About Predicates in One Variable
After doing the preceding exercises, you may be thinking of possible gen-
eral relationships, involving equivalence or implication, that might exist
between various quantified compound predicates. Some were hinted at
rather directly in the exercises (e.g., Exercises 5 and 8). Example 1 pre-
sents another approach to one of the pairs of statements you might have
compared in Exercise 8.EXAMPLE (^1) Describe in terms of truth sets the conditions under which
the quantified compound predicates (Vx)(-p(x)) and - [(3x)(p(x))] are
true.
Solution By Definition 2(a), Article 3.1, the truth set of --p(x) is P'.
By Definition l(a), Article 3.2, the universally quantified statement
(Vx)(-p(x)) is true if and only if P' = U. On the other hand, by Defini-
tion 2(a), Article 2.1, - [(3x)(p(x))] is true if and only if (3x)Mx)) is false.
This in turn [by Definition l(b), Article 3.21 is the case if and only if it is
false that P # @, that is, if and only if the statement P = is true. 0
Example 1 indicates that a statement of the form "for every x, not p(x)"
is true precisely when P' = U, while the corresponding statement "it is not
the case that there exists x for which p(x)" is true precisely when P = @.
Looking at these two conditions about truth sets, we see intuitively that
they - are logically equivalent, that is, either both true or both false (i.e.,
P' = U if and only if P # a). Accepting this unproved statement about sets
as true (recall the remarks in the last several paragraphs of Article 3.1),
we must conclude, by transitivity of the biconditional, that (Vx)(-p(x))
// and - [3x)(p(x))] are logically equivalent statements for any propositional
function p(x). This means that, for a given predicate Ax), either both are
true or both are false. This conclusion is consistent with results you should
have gotten in Eiercises 6 and 8, Article 3.2.
Several other theorems of the predicate calculus, involving equivalence
between pairs of quantified propositional functions of one variable, can be
arrived at in a similar manner. In Theorem 1 we list beside each statement
about quantified predicates a corresponding assertion about truth sets.
Note, in each case, that the latter statement about sets, although unproved
as yet, is intuitively believable. For further evidence to support the con-
clusions of Theorem 1, review your answers to Exercises 3(a, b, g, h) and
6, Article 3.2.