Test Bank For Discrete Mathematics with Application 4th Edition

Test Bank For Discrete Mathematics with Application 4th Edition

ISBN-10: 8131533026, ISBN-13: 978-8131533024

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.
11. Define a relation R from R to R as follows: For all (x, y) ∈ R × R,(x, y) ∈ R if, and only if,
x = y
2 + 1.
(a) Is (2, 5) ∈ R? Is (5, 2) ∈ R? Is (−3) R 10? Is 10 R (−3)?]
(b) Draw the graph of R in the Cartesian plane.
(c) Is R a function from A to B? Explain.
12. Let A = {1, 2, 3, 4} and B = {a, b, c}. Define a function G: A → B as follows:
G = {(1, b),(2, c),(3, b),(4, c)}.
(a) Find G(2).
(b) Draw an arrow diagram for G.
13. Define functions F and G from R to R by the following formulas:
F(x) = (x + 1)(x − 3) and G(x) = (x − 2)2 − 7.
Does F = G? Explain.