86 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3
Solution We first calculate P and Q and express them in interval notation.
The absolute value inequality lx 1 I 1 is equivalent to the two inequal-
ities -1 5 x 5 1 so that P = [-I, I], whereas lx - 11 < 1 may be
expressed - 1 < x - 1 < 1, so that 0 < x < 2 and Q = (0,2). By Defi-
nition 2(iii), the truth set of p(x) A q(x) is P n Q = (0, 11. By Definition
3(d), the truth set of p(x) -+ q(x) is P' u Q = (-a, - 1) u (0, oo).
Let us now test the reasonableness of these results. We consider first
p(x) A q(x). Let a = 0. Since 0 $ (0, 11 = P n Q, we expect that p(0) A q(0)
is false. This is so since q(0) is false (i.e., x = 0 does not satisfy the
inequality lx - 1 ( < 1). Check for yourself that a = - 5 and a = 2 result
in a truth value of "false" for p(a) A q(a). On the other hand, if we let
a = t, both p($) and q(*) are true (Why?) so that p(+)~ q($) is true
(Why?). This is no surprise since E (0, 11, the truth set of p(x) A q(x).
Try a = 1 as another example.
Next, we consider p(x) + q(x) a more difficult case for our intuition
to handle. We proceed mechanically, however. Let a = - 1; since
- 1 4 (- oo, - 1) u (0, a), we expect that p(- 1) -+ q(- 1) should be
false. This is indeed the case, since - 1 E [- 1, 11 = P [the truth set of
fix)] so that p( - 1) is true, whereas q(- 1) is false [since - 1 $ (0,2) = Q].
Recall that T -+ F is the only case in the truth table for -+ which yields
the truth value F. On the other hand, if we let a = 3, we find that p(3) +
q(3) is true since p(3) is false (3 4 P = 3 $ Q =
(0,2) is false [3 $ Q =
(0,2)]. This result is consistent with our calculation of the truth set of
p(x) -+ q(x) since 3 E(-a, -1) u (0, a) = P' u Q. Try a = f, a = $,
and a = -f for yourself. In each case, before determining the truth
value of fia) -* q(a) directly, make a prediction based on our calculation
of the truth set.
As we conclude this article, let us briefly discuss the logical direction
of material in this chapter. Our main goals here are the "theorems of
the predicate calculus" in Articles 3 and 4, especially Theorems 1 and 2,
Article 3.3, and Theorems 1 and 3, Article 3.4. These are crucially im-
portant principles of reasoning. Any serious student of mathematics at
- the junior-senior level must have at least general familiarity with them,
although a detailed understanding is preferable.
For a text at this level, formal proofs of theorems of the predicate
calculus are omitted. But although we present these theorems without
formal proof, we do not present them in a vacuum. Specifically, through
development of the notion of truth set in this article and in Article 3.2,
we will be able to justify a number of theorems of the predicate calculus
by means of corresponding theorems of set theory. The latter, of course,
have not been formally proved, but, based on the "intuitive feel" for sets
acquired in Chapter 1, you should be easily able to recognize them as
true. A danger in this approach is that we may seem to be using "circular"
reasoning, since in later chapters we will use principles of logic to prove
