Bridge to Abstract Mathematics: Mathematical Proof and Structures

3.1 BASIC CONCEPTS OF THE PREDICATE CALCULUS 87

theorems of set theory. You should understand, then, that our approach

is to accept theorems of the predicate calculus without proof. The pair-

ing of unproved theorems of set theory with theorems of logic in Articles

3.3 and 3.4 should not be viewed as an attempt to prove the latter, but

only as confirmation of their plausibility.

Exercises

In Exercises 1 through 4, let U = (1,2, 3,... ,9, 10). Let propositional functions

fix), q(x), r(x), and s(x) be defined on U by p(x): x 2 3, q(x): x 5 7, r(x):

x > 3, and s(x): x # 3.

1. Use the roster method to describe the truth sets P, Q, R, and S explicitly.


2. Use Definitions 2 and 3 to find the truth sets of the following compound prop.
   ositional functions:


3. (a) What would you expect to be the truth set of each of the following com-
   pound open sentences:

(b) Compute each of these truth sets directly from Definitions 2 and 3 [relative
-- to the specific predicates dx), q(x), r(x), and s(x) given before Exercise 11 and
compare the results with your expectations from (a).

4. (a) Calculate tfie truth sets of the following compound open sentences:

(b) What do the results in (a) suggest about the negation of the five connectives

in the context of propositional functions, compared to that of propositions, our

context in Article 2.3?