364 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES
(true), q: September is the name of a month (true), r: Wednesday is the name
of a month (false); in (f ), -(p A q A r) is true; in (g), - p A - q A - r is false.Article 2.2
- (a) (iu) I pass the course and (still) do not make the dean's list. (vi) If I
pass the course, then I make the dean's list. (b) (ii) q v -r (-q -+ -r is
also a correct representation) (iu) q -+ p (-p -+ - q is also a correct
representation). - (c) p: 2 = 5 (false), q: 4 + 5 = 9 (true), r: 52 = 25 (true); (- p v q) -+ - r is false.
(f) (h) p: S"_, sin x dx = 0 (true), q: dldx(2") = x2"- ' (false), r: In 6 = (In 2)(ln 3)
(false); in (f), (p A -q) -+ r is false; in (h), p -+ (r -+ q) is true.
Article 2.3
- (a) (i) converse: r -+ (p A q); contrapositive: -r -+ (-p v -q); inverse:
( - p v - q) -+ - r. (b) (iii) inverse (uii) original. (d) (ii) converse: If
sin (43) = 4, then 2 < 4 and 5 + 5 = 10; inverse: If either 2 2 4 or 5 + 5 # 10,
then sin (7113) # 4; contrapositive: If sin (43) # 4, then either 2 2 4 or
5 + 5 # 10; negation: 2 < 4 and 5 + 5 = 10, but sin (7113) # 3. - (b) -q-+ -r (e)(r-+q)-+p.
- (b) (i) In (a'), p is stronger than q, due to its stronger conclusion; in (b'), p is
stronger than q, due to its weaker hypothesis. (ii) In (a'), p and q are both
true; in (b'), p is false and q is true. (iii) Our answers to (ii) are consistent
with our answers to (i). The fact that p is stronger than q, in (a') and (b')
excludes only the possibility "p true-q false." - (b) (ii) -pv -qvr.
Article 2.4
- p: I keep my job, q: Smith is retained, r: You recommend Smith's firing. The
argument has the form [( - q -+ - p) A ( - q -+ r)] -+ ( - r -+ p), an invalid
argument (consider the case: r false, p false, and q true).
8. A: p -+ q is a tautology, B: q + p is a tautology, C: p - q is a tautology. The
argument has the form [(A -+ (B o C)) A (A A - C)] -+ - B, a valid argument.
Article 3.1
- (g) (4,563 7) (k) U (1) 0 (4 U.
- (a) (i) both equal (1, 2, 3,8,9, 10).