3.3 THEOREMS ABOUT PREDICATES IN ONE VARIABLE 99
(c) The statement form (Vx)(p(x) v q(x)) from Theorem 2(a) asserts, when true, that
for every substitution of an object a from a universal set U, either statement p(a)
or statement q(a) is true. Express in your own words the "extra" condition that
must hold if the stronger statement form (Vx)(p(x)) v (Vx)(q(x)) is also to be true.
(d) The statement form (3x)(p(x)) ~(3x)(q(x)) from Theorem 2(b) asserts, when
true, that there is some substitution of an object a from U such that Ha) is true
and there is some substitution of an object b from U such that q(b) is true. What
extra condition must hold in order that the stronger statement form (3x)(p(x) A
q(x)) also be true?
- (a) Give an example of propositional functions p(x) and q(x) over a nonempty
domain U for which (Vx)(p(x) v q(x)) is true and (Vx)(p(x)) v (Vx)(q(x)) is false,
whereas (3x)(p(x) A q(x)) and (3x)(p(x)) A (3x)(q(x)) both have the same truth
value. (Note: This combination did not occur in any of the examples (i) through
(vi) of Exercise 3, Article 3.2.)
(b) Is it possible to find p(x) and q(x) in part (a) so that (3x)(p(x) A q(x)) and
(3x)(p(x)) A (3x)(q(x)) are both false?
- In Example 3 the negation of (Vx)(p(x) -, q(x)) was expressed in the equivalent
form (3x)(p(x) A - q(x)). Note that the "not" connective does not occur as a "main
connective"; that is, it does not modify any compound proposition or compound
predicate in the latter form. Use a similar approach to express the negation of each
of the following statement forms in a form in which "not" does not appear as a
main connective:
- Express the negation of each of the following statement forms in a form that
does not employ the connective "not" as a main connective:
- In each of (a) through (e), describe precisely what must be proved in order
disprove the given (false) statement:
For every function f (with domain and range both equal to R), iff '(0) = 0,
then f has a relative maximum or minimum at x = 0.
For every square matrix A, if A is upper triangular, then A is diagonal.
For every curve C in R x R, if C is symmetric with respect to the x-axis,
then C is symmetric with respect to the origin.
For every function f, iff has a relative maximum or minimum at x = 0, then
f'(0) = 0.
For every function f, iff is defined at x = 0 and if both lim,+,- f(x) and
lim,,,, f (x) exist, then f is continuous at x = 0.
For every group G, if G is abelian, then G is cyclic.
For every subset S of a metric space X, S is compact if and only if S is closed and
bounded.
