74 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2- Let p, q, and r represent statements defined as follows:
p: lines x and y lie in the same plane.
q: lines x and y are parallel.
r: lines x and y have no points in common.
Express symbolically as compound statement forms in the letters p, q, and r.
If x and y are parallel, then they have no points in common.
Lines x and y intersect unless they are parallel.
Lines x and y have no points in common, but they are not parallel.
The statement "lines x and y are parallel" is stronger than the statement
"lines x and y have no points in common."
In order for x and y having no points in common to imply that x and y are
parallel, it is necessary that x and y lie in the same plane.
Whenever x and y are parallel, then x and y lie in the same plane.
Lines x and y intersect unless they are parallel or don't lie in the same plane.
Either x and y are parallel or x and y have points in common, but not both. - According to (v) of Theorem 1, two statement forms are equivalent if and only
if their negations are equivalent. Use this fact, together with the other parts of
Theorem 1 [except (q) and (u)] to argue the equivalence of the statement forms:
(a) (p v q) + r and (p + r) A (q -.+ r) [(q) of Theorem^11
(b) (PA q) + r and P + (q + [(u) of Theorem 11 - (a) Give three examples of well-known corollaries to the mean value theorem
of elementary calculus.
(b) In each of parts (a') through (g'), one of the given compound statements p or
q is formally stronger than the other; in fact, either p + q or q + p is an instance
of a tautology.
(i) Determine which of the two is stronger in each case.
(ii) Based on your previous mathematical experience, label each of the follow-
ing 14 statements as true or false [when each is preceded by the appropriate
number of occurrences of the universal quantifier for every. For example,
both sentences in (d') should be preceded by for every f and for every a].
(iii) Check that your answers in (ii) are consistent with your conclusions in (i).
*(a1) p: If 0 < Ix - a1 < /?, then x # a and a - /? < x < a + /?.
q: If 0 < Ix - a1 < /?, then a - /? < x < a + /3.
(b') p: If a, b, and x are real numbers such that ax = bx, then a = b.
q: If a, b, and x are real numbers such that ax = bx and x # 0, then a = b.
(c') p: Iff has a relative maximum at a and f is differentiable at a, then f '(a) = 0.
q: Iff has relative maximum at a, then f'(a) = 0.
(d') p: Iff is differentiable at a, then f is continuous at a.
q: Iff is differentiable at a and f has a relative maximum at a, then f is
continuous at a.
(e') p: Iff is defined at a and lirn,,, f(x) exists and equals f(a), then f is con-
tinuous at a.
q: Iff is defined at a and lim,,, f(x) exists, then f is continuous at a.
(f') p: If a, b, and p are integers, if p is prime, and p divides the product ab of a
and b, then either p divides a or p divides b.