SOLUTIONS MANUAL - Section 1.1 2.{2,4} 3.{7,10} 5.{2,3,5,6,8,9} 6....

SOLUTIONS MANUAL

DISCRETE MATHEMATICS

EI

GHTH EDI TION

Richard Johnsonbaugh 1 / 4

Solutions to Selected Exercises Section 1.1

2.{2,4} 3.{7,10} 5.{2,3,5,6,8,9} 6.{1,3,5,7,9,10}

8.A 9.∅ 11.B 12.{1,4} 14.{1}

15.{2,3,4,5,6,7,8,9,10} 18.{n∈Z + |n≥6} 19.{2n−1|n∈Z + } 21.{n∈Z + |n≤5 orn= 2m, m≥3} 22.{2n|n≥3} 24.{1,3,5} 25.{n∈Z + |n≤5 orn= 2m+ 1, m≥3} 27.{n∈Z + |n≥6 orn= 2 orn= 4}

29. 1 30. 3

• We find thatB={2,3}. SinceAandBhave the same elements, they are equal.
• Letx∈A. Thenx= 1,2,3. Ifx= 1, since 1∈Z

+ and 1 2 <10, thenx∈B. Ifx= 2, since 2∈Z + and 2 2 <10, thenx∈B. Ifx= 3, since 3∈Z + and 3 2 <10, thenx∈B. Thus ifx∈A, thenx∈B.Now suppose thatx∈B. Thenx∈Z + andx 2 <10. Ifx≥4, thenx 2 >10 and, for these values ofx, x /∈B. Thereforex= 1,2,3. For each of these values,x 2 <10 andxis indeed inB. Also, for each of the valuesx= 1,2,3,x∈A. Thus ifx∈B, thenx∈A. ThereforeA=B.

• Since (−1)

3

−2(−1)

2 −(−1) + 2 = 0,−1∈B. Since−1/∈A,A6=B.

• Since 3

2 −1>3, 3/∈B. Since 3∈A,A6=B. 41. Equal 42. Not equal

• Letx∈A. Thenx= 1,2. Ifx= 1,

x 3 −6x 2

• 11x= 1

3

−6∙1

2

+ 11∙1 = 6.

Thusx∈B. Ifx= 2, x 3 −6x 2

• 11x= 2

3

−6∙2

2

+ 11∙2 = 6.

Againx∈B. ThereforeA⊆B.

• Letx∈A. Thenx= (1,1) orx= (1,2). In either case,x∈B. ThereforeA⊆B.
• Since (−1)

3

−2(−1)

2 −(−1) + 2 = 0,−1∈A. However,−1/∈B. ThereforeAis not a subset ofB.

• Consider 4, which is inA. If 4∈B, then 4∈Aand 4 +m= 8 for somem∈C. However, the only value

ofmfor which 4 +m= 8 ism= 4 and 4/∈C. Therefore 4/∈B. Since 4∈Aand 4/∈B,Ais not a subset ofB.Copyrightc2018 Pearson Education, Inc. 2 / 4

2SOLUTIONS

53.U A B 54.U A B 56.U A B C 57.U A B C 59.U A B C

62. 32 63. 105 65. 51

• Suppose thatnstudents are taking both a mathematics course and a computerscience course. Then

4nstudents are taking a mathematics course, but not a computerscience course, and 7nstudents are taking a computer science course, but not a mathematics course. The following Venn diagram depicts

the situation:

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SOLUTIONS3

&% '$ &% '$ 4n n 7n MathCompSci Thus, the total number of students is 4n+n+ 7n= 12n.The proportion taking a mathematics course is 5n 12n = 5 12 , which is greater than one-third.

69.{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} 70.{(1,1),(1,2),(2,1),(2,2)} 73.{(1, a, a),(2, a, a)}

74.{(1,1,1),(1,2,1),(2,1,1),(2,2,1),(1,1,2),(1,2,2),(2,1,2),(2,2,2)}

• Vertical lines (parallel) spaced one unit apart extending infinitely to the left and right.
• Consider all points on a horizontal line one unit apart. Now copy these points by moving the horizontal

linenunits straight up and straight down for all integersn >0. The set of all points obtained in this way is the setZ×Z.

• Ordinary 3-space
• Take the lines described in the instructions for this setof exercises and copy them by movingnunits out

and back for alln >0. The set of all points obtained in this way is the setR×Z×Z.

84.{1,2}

{1},{2}

85.{a, b, c} {a, b},{c} {a, c},{b} {b, c},{a} {a},{b},{c}

• False 89. True 91. False 92. True

94.∅,{a},{b},{c},{d},{a, b},{a, c},{a, d},{b, c},{b, d},{c, d},{a, b, c},{a, b, d}, {a, c, d},{b, c, d},{a, b, c, d}. All except{a, b, c, d}are proper subsets.

• 2

10

= 1024; 2

10

−1 = 1023 98. B⊆A 99.A=U

• The symmetric difference of two sets consists of the elements in one or the other but not both.

103.A4A=∅,A4A=U,U4A=A,∅4A=A

• The set of primes

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