Page 36 - Discrete Mathematics and Its Applications

1.1 Propositional Logic 15

24. Write each of these statements in the form “if p, then q” c) q ∨ p ∨¬s ∨¬r ∨¬t ∨ u
in English. [Hint: Refer to the list of common ways to ex- d) (p ∧ r ∧ t) ↔ (q ∧ t)
press conditional statements provided in this section.] 30. How many rows appear in a truth table for each of these
a) I will remember to send you the address only if you compound propositions?
send me an e-mail message. a) (q →¬p) ∨ (¬p →¬q)
b) To be a citizen of this country, it is sufficient that you b) (p ∨¬t) ∧ (p ∨¬s)
were born in the United States.
c) If you keep your textbook, it will be a useful reference c) (p → r) ∨ (¬s →¬t) ∨ (¬u → v)
in your future courses. d) (p ∧ r ∧ s) ∨ (q ∧ t) ∨ (r ∧¬t)
d) The RedWings will win the Stanley Cup if their goalie 31. Construct a truth table for each of these compound propo-
plays well. sitions.
e) That you get the job implies that you had the best a) p ∧¬p b) p ∨¬p
credentials. c) (p ∨¬q) → q d) (p ∨ q) → (p ∧ q)
f) The beach erodes whenever there is a storm. e) (p → q) ↔ (¬q →¬p)
g) It is necessary to have a valid password to log on to
the server. f) (p → q) → (q → p)
h) Youwillreachthesummitunlessyoubeginyourclimb 32. Construct a truth table for each of these compound propo-
too late. sitions.
25. Write each of these propositions in the form “p if and a) p →¬p b) p ↔¬p
only if q” in English. c) p ⊕ (p ∨ q) d) (p ∧ q) → (p ∨ q)
a) If it is hot outside you buy an ice cream cone, and if e) (q →¬p) ↔ (p ↔ q)
you buy an ice cream cone it is hot outside. f) (p ↔ q) ⊕ (p ↔¬q)
b) For you to win the contest it is necessary and sufficient 33. Construct a truth table for each of these compound propo-
that you have the only winning ticket. sitions.
c) You get promoted only if you have connections, and a) (p ∨ q) → (p ⊕ q) b) (p ⊕ q) → (p ∧ q)
you have connections only if you get promoted.
d) If you watch television your mind will decay, and con- c) (p ∨ q) ⊕ (p ∧ q) d) (p ↔ q) ⊕ (¬p ↔ q)
versely. e) (p ↔ q) ⊕ (¬p ↔¬r)
e) The trains run late on exactly those days when I take f) (p ⊕ q) → (p ⊕¬q)
it. 34. Construct a truth table for each of these compound propo-
26. Write each of these propositions in the form “p if and sitions.
only if q” in English. a) p ⊕ p b) p ⊕¬p
a) For you to get an A in this course, it is necessary and c) p ⊕¬q d) ¬p ⊕¬q
sufficient that you learn how to solve discrete mathe- e) (p ⊕ q) ∨ (p ⊕¬q) f) (p ⊕ q) ∧ (p ⊕¬q)
matics problems. 35. Construct a truth table for each of these compound propo-
b) If you read the newspaper every day, you will be in- sitions.
formed, and conversely.
a) p →¬q b) ¬p ↔ q
c) It rains if it is a weekend day, and it is a weekend day
c) (p → q) ∨ (¬p → q) d) (p → q) ∧ (¬p → q)
if it rains.
d) You can see the wizard only if the wizard is not in, e) (p ↔ q) ∨ (¬p ↔ q)
and the wizard is not in only if you can see him. f) (¬p ↔¬q) ↔ (p ↔ q)
27. State the converse, contrapositive, and inverse of each of 36. Construct a truth table for each of these compound propo-
these conditional statements. sitions.
a) If it snows today, I will ski tomorrow. a) (p ∨ q) ∨ r b) (p ∨ q) ∧ r
b) I come to class whenever there is going to be a quiz. c) (p ∧ q) ∨ r d) (p ∧ q) ∧ r
c) A positive integer is a prime only if it has no divisors e) (p ∨ q) ∧¬r f) (p ∧ q) ∨¬r
other than 1 and itself.
37. Construct a truth table for each of these compound propo-
28. State the converse, contrapositive, and inverse of each of sitions.
these conditional statements.
a) p → (¬q ∨ r)
a) If it snows tonight, then I will stay at home. b) ¬p → (q → r)
b) I go to the beach whenever it is a sunny summer day. c) (p → q) ∨ (¬p → r)
c) When I stay up late, it is necessary that I sleep until
noon. d) (p → q) ∧ (¬p → r)
e) (p ↔ q) ∨ (¬q ↔ r)
29. How many rows appear in a truth table for each of these
compound propositions? f) (¬p ↔¬q) ↔ (q ↔ r)
a) p →¬p 38. Construct a truth table for ((p → q) → r) → s.
b) (p ∨¬r) ∧ (q ∨¬s) 39. Construct a truth table for (p ↔ q) ↔ (r ↔ s).