# Test Bank For Discrete Mathematics With Applications, 5th Edition

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Discrete Mathematics with Applications, 5th 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
(b) For any
,
s such that s is
.
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 satis๏ฌes the property that given any positive integer
n, m is
.
7. (a) Write in words how to read the following out loud {n โ Z | n is a factor of 9}.
(b) Use the set-roster notation to indicate the elements in the set.
8. (a) Is {5} โ {1, 3, 5}?
(b) Is {5} โ {1, 3, 5}?
(c) Is {5} โ {{1}, {3}, {5}}?
(d) Is {5} โ {{1}, {3}, {5}}?
9. Let A = {a, b, c} and B = {u, v}. Write a. A ร B and b. B ร A.
10. Let A = {3, 5, 7} and B = {15, 16, 17, 18}, and de๏ฌne a relation R from A to B as follows: For
all (x, y) โ A ร B,
y
(x, y) โ R โ
is an integer.
x
(a) Is 3 R 15? Is 3 R 16? Is (7, 17) โ R? Is (3, 18) โ 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. De๏ฌne a relation R from R to R as follows: For all (x, y) โ R ร R, (x, y) โ R if, and only if,
x = y 2 + 1.
(a) Is (2, 5) โ R? Is (5, 2) โ R? Is (โ3) R 10? Is 10 R (โ3)?
(b) Draw the graph of R in the Cartesian plane.
(c) Is R a function from R to R? Explain.
12. Let A = {1, 2, 3, 4} and B = {a, b, c}. De๏ฌne a function G: A โ B as follows:
G = {(1, b), (2, c), (3, b), (4, c)}.
(a) Find G(2).
(b) Draw an arrow diagram for G.
13. De๏ฌne functions F and G from R to R by the following formulas:
F (x) = (x + 1)(x โ 3) and G(x) = (x โ 2)2 โ 7.
Does F = G? Explain.
Chapter 2
1. Which of the following is a negation for โJim is inside and Jan is at the pool.โ
(a) Jim is inside or Jan is not at the pool.
(b) Jim is inside or Jan is at the pool.
(c) Jim is not inside or Jan is at the pool.
(d) Jim is not inside and Jan is not at the pool.
(e) Jim is not inside or Jan is not at the pool.
2
2. Which of the following is a negation for โJim has grown or Joan has shrunk.โ
(a) Jim has grown or Joan has shrunk.
(b) Jim has grown or Joan has not shrunk.
(c) Jim has not grown or Joan has not shrunk.
(d) Jim has grown and Joan has shrunk.
(e) Jim has not grown and Joan has not shrunk.
(f) Jim has grown and Joan has not shrunk.
3. Write a negation for each of the following statements:
(a) The variable S is undeclared and the data are out of order.
(b) The variable S is undeclared or the data are out of order.
(c) If Al was with Bob on the ๏ฌrst, then Al is innocent.
(d) โ5 โค x x or x โฅ 2
4. The statement forms are not logically equivalent.
Truth table:
p
T
T
F
F
q
T
F
T
F
โผp
F
F
T
T
pโจq
T
T
T
F
โผpโง q
F
F
T
F
pโจq โp
T
T
F
T
p โจ (โผ p โง q)
T
T
T
F
Explanation: The truth table shows that p โจ q โ p and p โจ (โผ p โง q) have di๏ฌerent truth values in
rows 3 and 4, i.e, when p is false. Therefore p โจ q โ p and p โจ (โผ p โง q) are not logically equivalent.
5. Sample answers:
Two statement forms are logically equivalent if, and only if, they always have the same truth values.
Or: Two statement forms are logically equivalent if, and only if, no matter what statements are
substituted in a consistent way for their statement variables the resulting statements have the same
truth value.
6. Solution 1: The given statements are not logically equivalent. Let p be โSam bought it at Crown
Books,โ and q be โSam didnโt pay full price.โ Then the two statements have the following form:
p โ q and
pโจ โผ q.
The truth tables for these statement forms are
p
T
T
F
F
q
T
F
T
F
โผq
F
T
F
T
pโq
T
F
T
T
2
pโจ โผ q
T
T
F
T
Rows 2 and 3 of the table show that p โ q and pโจ โผ q do not always have the same truth values, and
so p โ q ฬธโก pโจ โผ q.
Solution 2 : The given statements are not logically equivalent. Let p be โSam bought it at Crown
Books,โ and q be โSam paid full price.โ Then the two statements have the following form:
p โโผ q and
p โจ q.
The truth tables for these statement forms are
p
T
T
F
F
q
T
F
T
F
โผq
F
T
F
T
p โโผ q
F
T
T
T
pโจq
T
T
T
F
Rows 1 and 4 of the table show that p โโผ q and p โจ q do not always have the same truth values, and
so p โโผ q ฬธโก p โจ q.
7. The given statements are not logically equivalent. Let p be โSam is out of Schlitz,โ and q be โSam is
out of beer.โ Then the two statements have the following form:
p โ q and
โผ qโจ โผ p.
The truth tables for these statement forms are
p
T
T
F
F
q
T
F
T
F
โผp
F
F
T
T
โผq
F
T
F
T
pโq
T
F
T
T
โผ pโจ โผ q
F
T
T
T
The table shows that p โ q and โผ p โจ โผ q sometimes have opposite truth values (shown in rows 1 and
2), and so p โ q ฬธโก โผ p โจ โผ q.
8. Converse: If Jose is Janโs cousin, then Ann is Janโs mother
Inverse: If Ann is not Janโs mother, then Jose is not Janโs cousin.
Contrapositive: If Jose is not Janโs cousin, then Ann is not Janโs mother.
9. Converse: If Liu is Sueโs cousin, then Ed is Sueโs father.
Inverse: If Ed is not Sueโs father, then Liu is not Sueโs cousin
Contrapositive: If Liu is not Sueโs cousin, then Ed is not Sueโs father.
10. Converse: If Jim is Tomโs grandfather, then then Al is Tomโs cousin.
Inverse: If Al is not Tomโs cousin, then Jim is not Tomโs grandfather
Contrapositive: If Jim is not Tomโs grandfather, then then Al is not Tomโs cousin.
11. If someone does not get an answer of 10 for problem 16, then the person will not have solved problem
16 correctly.
Or: If someone solves problem 16 correctly, then the person got an answer of 10.
12. Sample answers:
For a form of argument to be valid means that no matter what statements are substituted for its
statement variables, if the resulting premises are all true, then the conclusion is also true.
Or: For a form of argument to be valid means that no matter what statements are substituted for its
statement variables, it is impossible for all the premises to be true at the same time that the conclusion
is false.
Or: For a form of argument to be valid means that no matter what statements are substituted for its
statement variables, it is impossible for conclusion to be false if all the premises are true.
3
13. The given form of argument is invalid.
premises
z
p
T
T
F
F
q
T
F
T
F
โผp
F
F
T
T
โผq
F
T
F
T
}|
p โโผ q
F
T
T
T
{
q โโผ p
F
T
T
T
conclusion
z
}|
{
pโจq
T
T
T
F
Row 4 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion.
14. The given form of argument is invalid.
premises
z
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
โผq
F
F
T
T
F
F
T
T
pโงโผq
F
F
T
T
F
F
F
F
}|
pโงโผqโr
T
T
T
F
T
T
T
T
{
pโจq
T
T
T
T
T
T
F
F
qโp
T
T
T
T
F
F
T
T
conclusion
z }|{
r
T
F
T
F
T
F
T
F
Row 2 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion.
15. Let p be โHugo is a physics major,โ q be โHugo is a math major,โ and r be โHugo needs to take
calculus.โ Then the given argument has the following form:
Therefore
Truth table:
premises
z
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
pโจq
F
F
T
T
T
T
F
F
pโจq โr
rโจq
p โจ q.
}|
pโจq โr
T
F
T
F
T
F
T
T
rโจq
T
T
T
F
T
T
T
F
{
conclusion
z }|{
pโจq
T
T
T
T
T
T
F
F
Row 7 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion. Therefore, the given argument is invalid.
16. Let p be โ12 divides 709,438,โ q be โ3 divides 709,438,โ and r be โThe sum of the digits of 709,438 is
divisible by 9.โ Then the given argument has the following form:
Therefore
4
pโq
rโq
โผr
โผ p.
Truth table:
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
โผq
F
F
T
T
F
F
T
T
pโงโผq
F
F
T
T
F
F
F
F
z
premises
pโq
T
T
F
F
T
T
T
T
rโq
T
T
F
T
T
T
F
T
}|
{
โผr
F
T
F
T
F
T
F
T
conclusion
z }|{
โผp
F
F
F
F
T
T
T
T
Row 2 of the truth table shows that it is possible for an argument of this form to have true premises
and a false conclusion. Therefore, the given argument is invalid.
17. The argument has the form
Therefore
pโq
โผq
โผ p,
which is valid by modus tollens (and the fact that the negation of โ17 is not a divisor of 54,587โ is โ17
is a divisor of 54,587โ).
18. The argument has the form
Therefore
pโq
q
p,
which is invalid; it exhibits the converse error.
19. A and B are knights, and C is a knave.
Reasoning: A cannot be a knave because if A were a knave his statement would be true, which is
impossible for a knave. Hence A is a knight, and at least one of the three is a knave. That implies
that at most two of the three are knaves, which means that Bโs statement is true. Hence B is a knight.
Since at least one of the three is a knave and both A and B are knights, it follows that C is a knave.
20. a. S = 1
b. โผ (P โง Q)โง (Q โง R)
21. 1101012 = 1 ยท 25 + 1 ยท 24 + 1 ยท 22 + 1 ยท 20 = 32 + 16 + 4 + 1 = 5310
22. 7510 = 64 + 8 + 2 + 1 = 1 ยท 26 + 0 ยท 25 + 0 ยท 24 + 1 ยท 23 + 0 ยท 22 + 1 ยท 21 + 1 ยท 20 = 10010112 .
23. The following circuit corresponds to the given Boolean expression:
P
Q
AND
OR
NOT
AND
NOT
5
R
24. One circuit (among many) having the given input/output table is the following:
P
Q
NOT
OR
AND
S
R
NOT
AND
25.
+
101112
10112
1000102
26. 1001102 = 1 ยท 25 + 0 ยท 24 + 0 ยท 23 + 1 ยท 22 + 1 ยท 21 + 0 ยท 20 = 32 + 4 + 2 = 3810
27. 4910 = (32 + 16 + 1)10 = 001100012 โโ 11001110 โโ 11001111.
So the twoโs complement is 11001111.
Check: 28 โ 49 = 256 โ 49 = 207 and
110011112
=
=
=
1 ยท 27 + 1 ยท 26 + 0 ยท 25 + 0 ยท 24 + 1 ยท 23 + 1 ยท 22 + 1 ยท 21 + 1 ยท 20
128 + 64 + 8 + 4 + 2 + 1
207,
which agrees.
Chapter 3
1. โ valid argument x, if x has true premises, then x has a true conclusion.
2. a. โ odd integer n, n2 is odd.
b. โ integer n, if n is odd then n2 is odd.
c. โ an odd integer n such that n2 is not odd.
Or: โ an integer n such that n is odd and n2 is not odd.
3. โ rational number r, โ integers u and v such that r is the ratio of u to v.
Or: โ rational number r, โ integers u and v such that r = u/v.
4. โ even integer n that is greater than 2, โ prime numbers p and q such that n = p + q.
Or: โ even integer n, if n > 2 then โ prime numbers p and q such that n = p + q.
5. e
6. a
7. d
8. g
6

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