Discrete Mathematics with Applications, 4th Edition

by Susanna S. Epp

Test Bank Questions

Chapter 1

1. Fill in the blanks to rewrite the following statement with variables: Is there an integer with a

remainder of 1 when it is divided by 4 and a remainder of 3 when it is divided by 7?

(a) Is there an integer n such that n has ?

(b) Does there exist such that if n is divided by 4 the remainder is 1 and if ?

2. Fill in the blanks to rewrite the following statement with variables:

Given any positive real number, there is a positive real number that is smaller.

(a) Given any positive real number r, there is s such that s is .

(b) For any , such that s < r.

3. Rewrite the following statement less formally, without using variables:

There is an integer n such that 1/n is also an integer.

4. Fill in the blanks to rewrite the following statement:

For all objects T, if T is a triangle then T has three sides.

(a) All triangles .

(b) Every triangle .

(c) If an object is a triangle, then it .

(d) If T , then T .

(e) For all triangles T, .

5. Fill in the blanks to rewrite the following statement:

Every real number has an additive inverse.

(a) All real numbers .

(b) For any real number x, there is for x.

(c) For all real numbers x, there is real number y such that .

6. Fill in the blanks to rewrite the following statement:

There is a positive integer that is less than or equal to every positive integer.

(a) There is a positive integer m such that m is .

(b) There is a such that every positive integer.

(c) There is a positive integer m which satisfies the property that given any positive integer

n, m is .

7. (a) Write in words how to read the following out loud {n ∈ Z | n is a factor of 9}.

(b) Use the set-roster notation to indicate the elements in the set.

8. (a) Is {5} ∈ {1, 3, 5}?

(b) Is {5} ⊆ {1, 3, 5}?

(c) Is {5} ∈ {{1}, {3}, {5}}?

(d) Is {5} ⊆ {{1}, {3}, {5}}?

9. Let A = {a, b, c} and B = {u, v}. Write a. A × B and b. B × A.

10. Let A = {3, 5, 7} and B = {15, 16, 17, 18}, and define a relation R from A to B as follows: For

all (x, y) ∈ A × B,

(x, y) ∈ R ⇔

y

x

is an integer.

(a) Is 3 R 15? Is 3 R 16? Is (7, 17) ∈ R? Is (3, 18) ∈ R?

(b) Write R as a set of ordered pairs.

(c) Write the domain and co-domain of R.

(d) Draw an arrow diagram for R.

(e) Is R a function from A to B? Explain

Answers for Test Bank Questions: Chapters 1-4

Please use caution when using these answers. Small differences in wording, notation, or choice of examples

or counterexamples may be acceptable.

Chapter 1

1. a. a remainder of 1 when it is divided by 4 and a remainder of 3 when it is divided by 7

b. an integer n; n is divided by 7 the remainder is 3

2. a. a positive real number; smaller than r

b. positive real number r; there is a positive real number s

Fill in the blanks to rewrite the following statement with variables:

3. There is an integer whose reciprocal is also an integer.

4. a. have three sides

b. has three sides

c. has three sides

d. is a triangle; has three sides

e. T has three sides

5. a. have additive inverses

b. an additive inverse

c. y is an additive inverse for x

6. a. less than or equal to every positive integer

b. positive integer m; less than or equal to every positive integer

c. less than or equal to n

7. (a) The set of all integers n such that n is a factor of 9.

Or: The set of all elements n in Z such that n is a factor of 9.

Or: The set of all elements n in the set of all integers such that n is a factor of 9.

(b) {1, 3, 9}

8. (a) No

(b) Yes

(c) Yes

(d) No

9. a. {(a, u),(a, v),(b, u),(b, v),(c, u),(c, v)}

b. {(u, a),(v, a),(u, b),(v, b),(u, c),(v, c)}

10. a. Yes; No; No; Yes

b. {(3, 15),(3, 18),(5, 15)}

c. domain is {3, 5, 7}; co-domain is {15, 16, 17, 18}.

d. Draw an arrow diagram for R.

e. No: R fails both conditions for being a function from A to B. (1) Elements 5 and 7 in A are not

related to any elements in B, and (2) there is an element in A, namely 3, that is related to two different

elements in B, namely 15 and 18.

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