Page 347 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Appendix 1: Solutions to Selected Exercises 333
6. (a) Either make a truth table, or reason as follows:
(P → R) ∧ (Q → R) is equivalent to (¬P ∨ R) ∧ (¬Q ∨ R)
which is equivalent to (¬P ∧¬Q) ∨ R
which is equivalent to ¬(P ∨ Q) ∨ R
which is equivalent to (P ∨ Q) → R
(b) (P → R) ∨ (Q → R) is equivalent to (P ∧ Q) → R.
8. ¬(P →¬Q).

Chapter 2

Section 2.1

1. (a) ∀x[∃yF(x, y) → S(x)], where F(x, y) stands for “x has forgiven y,”
and S(x) stands for “x is a saint.”
(b) ¬∃x[C(x) ∧∀y(D(y) → S(x, y))], where C(x) stands for “x is in the
calculus class,” D(y) stands for “y is in the discrete math class,” and
S(x, y) stands for “x is smarter than y.”
(c) ∀x(¬(x = m) → L(x, m)), where L(x, y) stands for “x likes y,” and
m stands for Mary.
(d) ∃x(P(x) ∧ S( j, x)) ∧∃y(P(y) ∧ S(r, y)), where P(x) stands for “x is
a police officer,” S(x, y) stands for “x saw y,” j stands for Jane, and
r stands for Roger.
(e) ∃x(P(x) ∧ S( j, x) ∧ S(r, x)), where the letters have the same mean-
ings as in part (d).
4. (a) All unmarried men are unhappy.
(b) y is a sister of one of x’s parents; i.e., y is x’s blood aunt.
7. (a), (d), and (e) are true; (b), (c), and (f) are false.

Section 2.2

1. (a) ∃x[M(x) ∧∀y(F(x, y) →¬H(y))], where M(x) stands for “x is ma-
joring in math,” F(x, y) stands for “x and y are friends,” and H(y)
stands for “y needs help with his homework.” In English: There is a
math major all of whose friends don’t need help with their homework.
(b) ∃x∀y(R(x, y) →∃zL(y, z)), where R(x, y) stands for “x and y are
roommates” and L(y, z) stands for “y likes z.” In English: There is
someone all of whose roommates like at least one person.
(c) ∃x[(x ∈ A ∨ x ∈ B) ∧ (x /∈ C ∨ x ∈ D)].
2
(d) ∀x∃ y[y > x ∧∀z(z + 5z = y)].
4. Hint: Begin by replacing P(x) with ¬P(x) in the first quantifier negation
law, to get the fact that ¬∃x¬P(x) is equivalent to ∀x¬¬P(x).