Discrete Structures, Logic, and Computability, 4th Edition

Chapter 7

Section 7.1

1. a. [p(0, 0) $\wedge$ p(0, 1)] $\vee$ [p(1, 0) $\wedge$ p(1, 1)] .

2. a. $\forall$x q(x), where x $\in$ {0, 1}.

c. $\forall$ y p(x, y), where y $\in$ {0, 1}.

e. $\exists x$ p(x), where x is an odd natural number.

3. a. x is a term. Therefore, p(x) is a wff, and it follows that $\exists x$ p(x) and $\forall x$ p(x) are wffs. Thus, $\exists x$ p(x) $\to$ $\forall x$ p(x) is a wff.

4. It is illegal to have an atom, p(x) in this case, as an argument to a predicate.

5. a. The three occurrences of x, left to right, are free, bound, and bound. The four occurrences of y, left to right, are free, bound, bound, and free.

c. The three occurrences of x, left to right, are free, bound, and bound. Both occurrences of y are free.

6. $\forall x$ p(x, y, z) $\to$ $\exists z$ q(z).