94 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3- Use your true-false answers from Exercise 3 to compare certain pairs of statement
forms from that exercise, namely:
(A) (a) with (b) (compare the first two rows of the matrix from Exercise 3)
(B) (c) with (d)
(C) (e) with (f)
Are any general conclusions suggested by these comparisons? (Note: This exercise,
along with Exercise 8, anticipates Theorems 1 and 2 of the next article.)- In (i) through (iv) (of the list of pairs of open sentences preceding Exercise 3), use
Definition 1, in combination with Definition 2, Article 3.1 [part (a)] to label either
true or false each of the following statement forms. (As in Exercise 3, set up a
matrix of T's and F's, this time of shape 4 x 4.)
(a) (VX)( - h(x) 1 (b) - [(Vx)(h(x) 11
(c) (W( - h(x) (dl - C(Wh(x))l - In parts (a) through (d) of Exercise 6, translate the four symbolized forms into
English sentences corresponding to each of (i) through (iv). In each case (a total of
16), compare your answer of true or false in Exercise 6 with your intuitive judgment
of the truth or falsehood of your English translation. - Use your true-false answers from Exercise 6 for each of the four cases (i) through
(iv), to compare pairs of the statement forms (a) through (d) in Exercise 6. Are any
general conclusions suggested by these comparisons? - Recall, from Definition 1, Article 3.1, that propositional functions Ax) and q(x)
are equivalent over a common domain U if and only if P = Q.
(a) Write a quantified compound predicate involving arbitrary predicates p(x) and
q(x) that should, intuitively, be a true statement precisely when p(x) and q(x)
are equivalent.
(b) Write an equation involving the truth sets P and Q that must be satisfied if the
quantified statement you wrote in (a) is true.
*(c) Use the result of (b), together with Definition 1, Article 3.1, to state a possible
theorem of set theory suggested by the equivalence ("precisely when") implied
in (a).
I/
- Analogous to Definition 1, Article 3.1, we say, given propositional functions
p(x) and q(x) over a common domain U, that p(x) implies q(x) if and only if P c Q.
(a) Write a quantified compound predicate involving p(x) and q(x) that should,
intuitively, be a true statement precisely when Ax) implies q(x).
(b) Write an equation involving the truth sets P and Q that must be satisfied if the
quantified statement you wrote in (a) is true.
(c) Use the result of (b), together with Definition 1, to state a possible theorem
of set theory suggested by the equivalence implied in (a).
I 11. In the following list, (a) through (i), of propositional functions over the domain
of discourse U = R, find all instances of pairs that are either equivalent or in which
one implies the other:
(a) p(x): 1x1 5 1 (b) q(x): - 4 < x t 4