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# 偶数题答案--离散数学及其应用（英文版·第7版）.pdf

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Section 1.1 Propositional Logic 1

CHAPTER 1

The Foundations: Logic and Proofs

SECTION 1.1 Propositional Logic

2. Propositions must have clearly deﬁned truth values, so a proposition must be a declarative sentence with no

free variables.

a) This is not a proposition; it’s a command.

b) This is not a proposition; it’s a question.

c) This is a proposition that is false, as anyone who has been to Maine knows.

d) This is not a proposition; its truth value depends on the value of x.

e) This is a proposition that is false.

f) This is not a proposition; its truth value depends on the value of n.

4. a) Jennifer and Teja are not friends.

b) There are not 13 items in a baker’s dozen. (Alternatively: The number of items in a baker’s dozen is not

equal to 13.)

c) Abby sent fewer than 101 text messages yesterday. Alternatively, Abby sent at most 100 text messages

yesterday. Note: The ﬁrst printing of this edition incorrectly rendered this exercise with “every day” in

place of “yesterday.” That makes it a much harder problem, because the days are quantiﬁed, and quantiﬁed

propositions are not dealt with until a later section. It would be incorrect to say that the negation in that

case is “Abby sent at most 100 text messages every day.” Rather, a correct negation would be “There exists a

day on which Abby sent at most 100 text messages.” Saying “Abby did not send more than 100 text messages

every day” is somewhat ambiguous—do we mean ¬∀ or do we mean ∀¬?

d) 121 is not a perfect square.

6. a) True, because 288 > 256 and 288 > 128.

b) True, because C has 5 MP resolution compared to B’s 4 MP resolution. Note that only one of these

conditions needs to be met because of the word or .

c) False, because its resolution is not higher (all of the statements would have to be true for the conjunction

to be true).

d) False, because the hypothesis of this conditional statement is true and the conclusion is false.

e) False, because the ﬁrst part of this biconditional statement is false and the second part is true.

8. a) I did not buy a lottery ticket this week.

b) Either I bought a lottery ticket this week or [in the inclusive sense] I won the million dollar jackpot on

Friday.

c) If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday.

d) I bought a lottery ticket this week and I won the million dollar jackpot on Friday.

e) I bought a lottery ticket this week if and only if I won the million dollar jackpot on Friday.

f) If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot on Friday.

2 Chapter 1 The Foundations: Logic and Proofs

g) I did not buy a lottery ticket this week, and I did not win the million dollar jackpot on Friday.

h) Either I did not buy a lottery ticket this week, or else I did buy one and won the million dollar jackpot on

Friday.

10. a) The election is not decided.

b) The election is decided, or the votes have been counted.

c) The election is not decided, and the votes have been counted.

d) If the votes have been counted, then the election is decided.

e) If the votes have not been counted, then the election is not decided.

f) If the election is not decided, then the votes have not been counted.

g) The election is decided if and only if the votes have been counted.

h) Either the votes have not been counted, or else the election is not decided and the votes have been counted.

Note that we were able to incorporate the parentheses by using the words either and else.

12. a) If you have the ﬂu, then you miss the ﬁnal exam.

b) You do not miss the ﬁnal exam if and only if you pass the course.

c) If you miss the ﬁnal exam, then you do not pass the course.

d) You have the ﬂu, or miss the ﬁnal exam, or pass the course.

e) It is either the case that if you have the ﬂu then you do not pass the course or the case that if you miss

the ﬁnal exam then you do not pass the course (or both, it is understood).

f) Either you have the ﬂu and miss the ﬁnal exam, or you do not miss the ﬁnal exam and do pass the course.

14. a) r ∧¬q b) p ∧ q ∧ r c) r → p d) p ∧ ¬q ∧ r e) (p ∧ q) → r f) r ↔ (q ∨ p)

16. a) This is T ↔ T, which is true.

b) This is T ↔ F, which is false.

c) This is F ↔ F, which is true.

d) This is F ↔ T, which is false.

18. a) This is F → F, which is true.

b) This is F → F, which is true.

c) This is T → F, which is false.

d) This is T → T, which is true.

20. a) The employer making this request would be happy if the applicant knew both of these languages, so this

is clearly an inclusive or .

b) The restaurant would probably charge extra if the diner wanted both of these items, so this is an exclusive

or.

c) If a person happened to have both forms of identiﬁcation, so much the better, so this is clearly an inclusive

or.

d) This could be argued either way, but the inclusive interpretation seems more appropriate. This phrase

means that faculty members who do not publish papers in research journals are likely to be ﬁred from their

jobs during the probationary period. On the other hand, it may happen that they will be ﬁred even if they

do publish (for example, if their teaching is poor).

22. a) The necessary condition is the conclusion: If you get promoted, then you wash the boss’s car.

b) If the winds are from the south, then there will be a spring thaw.

Section 1.1 Propositional Logic 3

c) The suﬃcient condition is the hypothesis: If you bought the computer less than a year ago, then the

warranty is good.

d) If Willy cheats, then he gets caught.

e) The “only if” condition is the conclusion: If you access the website, then you must pay a subscription fee.

f) If you know the right people, then you will be elected.

g) If Carol is on a boat, then she gets seasick.

24. a) If I am to remember to send you the address, then you will have to send me an e-mail message. (This has

been slightly reworded so that the tenses make more sense.)

b) If you were born in the United States, then you are a citizen of this country.

c) If you keep your textbook, then it will be a useful reference in your future courses. (The word “then” is

understood in English, even if omitted.)

d) If their goaltender plays well, then the Red Wings will win the Stanley Cup.

e) If you get the job, then you had the best credentials.

f) If there is a storm, then the beach erodes.

g) If you log on to the server, then you have a valid password.

h) If you do not begin your climb too late, then you will reach the summit.

26. a) You will get an A in this course if and only if you learn how to solve discrete mathematics problems.

b) You will be informed if and only if you read the newspaper every day. (It sounds better in this order; it

would be logically equivalent to state this as “You read the newspaper every day if and only if you will be

informed.”)

c) It rains if and only if it is a weekend day.

d) You can see the wizard if and only if he is not in.

28. a) Converse: If I stay home, then it will snow tonight. Contrapositive: If I do not stay at home, then it will

not snow tonight. Inverse: If it does not snow tonight, then I will not stay home.

b) Converse: Whenever I go to the beach, it is a sunny summer day. Contrapositive: Whenever I do not go

to the beach, it is not a sunny summer day. Inverse: Whenever it is not a sunny day, I do not go to the beach.

c) Converse: If I sleep until noon, then I stayed up late. Contrapositive: If I do not sleep until noon, then I

did not stay up late. Inverse: If I don’t stay up late, then I don’t sleep until noon.

30. A truth table will need 2

n

rows if there are n variables.

a) 2

2

= 4 b) 2

3

= 8 c) 2

6

= 64 d) 2

5

= 32

32. To construct the truth table for a compound proposition, we work from the inside out. In each case, we will

show the intermediate steps. In part (d), for example, we ﬁrst construct the truth tables for p ∧ q and for

p ∨ q and combine them to get the truth table for (p ∧ q ) → (p ∨ q). For parts (a) and (b) we have the

following table (column three for part (a), column four for part (b)).

p ¬p p → ¬p p ↔ ¬p

T F F F

F T T F

For parts (c) and (d) we have the following table.

p q p ∨ q p ∧ q p ⊕ (p ∨ q) (p ∧ q) → (p ∨ q)

T T T T F T

T F T F F T

F T T F T T

F F F F F T

4 Chapter 1 The Foundations: Logic and Proofs

For part (e) we have the following table.

p q ¬p q → ¬p p ↔ q (q → ¬p) ↔ (p ↔ q)

T T F F T F

T F F T F F

F T T T F F

F F T T T T

For part (f ) we have the following table.

p q ¬q p ↔ q p ↔ ¬q (p ↔ q) ⊕ (p ↔ ¬q)

T T F T F T

T F T F T T

F T F F T T

F F T T F T

34. For parts (a) and (b) we have the following table (column two for part (a), column four for part (b)).

p p ⊕ p ¬p p ⊕ ¬p

T F F T

F F T T

For parts (c) and (d) we have the following table (columns ﬁve and six).

p q ¬p ¬q p ⊕ ¬q ¬p ⊕ ¬q

T T F F T F

T F F T F T

F T T F F T

F F T T T F

For parts (e) and (f) we have the following table (columns ﬁve and six). This time we have omitted the column

explicitly showing the negation of q . Note that the ﬁrst is a tautology and the second is a contradiction (see

deﬁnitions in Section 1.3).

p q p ⊕ q p ⊕ ¬q (p ⊕ q) ∨ (p ⊕ ¬q) (p ⊕ q) ∧ (p ⊕ ¬q)

T T F T T F

T F T F T F

F T T F T F

F F F T T F

36. For parts (a) and (b), we have

p q r p ∨ q (p ∨ q) ∨ r (p ∨ q) ∧ r

T T T T T T

T T F T T F

T F T T T T

T F F T T F

F T T T T T

F T F T T F

F F T F T F

F F F F F F

For parts (c) and (d), we have

Section 1.1 Propositional Logic 5

p q r p ∧ q (p ∧ q) ∨ r (p ∧ q) ∧ r

T T T T T T

T T F T T F

T F T F T F

T F F F F F

F T T F T F

F T F F F F

F F T F T F

F F F F F F

Finally, for parts (e) and (f ) we have

p q r ¬r p ∨ q (p ∨ q) ∧ ¬r p ∧ q (p ∧ q) ∨ ¬r

T T T F T F T T

T T F T T T T T

T F T F T F F F

T F F T T T F T

F T T F T F F F

F T F T T T F T

F F T F F F F F

F F F T F F F T

38. This time the truth table needs 2

4

= 16 rows.

p q r s p → q (p → q) → r ((p → q) → r) → s

T T T T T T T

T T T F T T F

T T F T T F T

T T F F T F T

T F T T F T T

T F T F F T F

T F F T F T T

T F F F F T F

F T T T T T T

F T T F T T F

F T F T T F T

F T F F T F T

F F T T T T T

F F T F T T F

F F F T T F T

F F F F T F T

40. This statement is true if and only if all three clauses, p ∨¬q , q ∨¬r, and r ∨¬p are true. Suppose p, q , and

r are all true. Because each clause has an unnegated variable, each clause is true. Similarly, if p, q , and r

are all false, then because each clause has a negated variable, each clause is true. On the other hand, if one of

the variables is true and the other two false, then the clause containing the negation of that variable will be

false, making the entire conjunction false; and similarly, if one of the variables is false and the other two true,

then the clause containing that variable unnegated will be false, again making the entire conjunction false.

42. a) Since the condition is true, the statement is executed, so x is incremented and now has the value 2.

b) Since the condition is false, the statement is not executed, so x is not incremented and now still has the

value 1.

c) Since the condition is true, the statement is executed, so x is incremented and now has the value 2.

d) Since the condition is false, the statement is not executed, so x is not incremented and now still has the

value 1.

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