1.1 Sequences ∙∙∙ 5
2006 Vasta & Fisher
1.1 Sequences – Exercises
Find the next term and classify each sequence as arithmetic, geometric, Fibonacci-like, or
none of these.
1. 3, 8, 13, 18, 23,
2. 3, 8, 11, 19, 30,
3. 4, 12, 36, 108, 324,
4. 4, 12, 20, 28, 36,
5. 1, 2, 4, 5, 7, 8,
6. 2, –2, 2, –2, 2, –2,
7. 3, –2, 1, –1, 0, –1,
8. 1, 2, 4, 7, 11, 16,
9. 1, 4, 9, 16, 25, 36,
10. 3, 6, 12, 24, 48, 96,
11. 16, 13, 10, 7, 4, 1,
12. –5, 4, –1, 3, 2, 5,
13. 1, 2, 3, 4, 5, 6,
14. 5, 7, 1, 3, –3, –1,
15. ,27
1,
9
1,
3
1,1,3,9
16. ,2
11,
2
7,2,
2
3,
2
1,1
17. 1, 2, 6, 13, 23, 36,
18. 1, 3, 6, 8, 16, 18, 36,
19. 2, 3, 5, 7, 11, 13,
20. 1, 2, 2, 4, 8, 32,
6 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.1 Sequences – Answers to Exercises
1. 28, arithmetic
2. 49, Fibonacci-like
3. 972, geometric
4. 44, arithmetic
5. 10, none of these
6. 2, geometric
7. –1, Fibonacci-like
8. 22, none of these
9. 49, none of these
10. 192, geometric
11. –2, arithmetic
12. 7, Fibonacci-like
13. 7, arithmetic
14. –7, none of these
15. 1/81, geometric
16. 9, Fibonacci-like
17. 52, none of these
18. 38, none of these
19. 17, none of these
20. 256, none of these
1.2 Pascal’s Triangle ∙∙∙ 11
2006 Vasta & Fisher
1.2 Pascal's Triangle – Exercises
Construct Pascal's Triangle up to row 9.
12 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.2 Pascal's Triangle – Answers to Exercises
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
16 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.3 Direct Routes – Exercises
How many direct routes are there from A to B?
1. 2.
3. 4.
5. 6.
A
B
A
B
A
B
A
B
A
B
A
B
G
1.3 Direct Routes ∙∙∙ 17
2006 Vasta & Fisher
7. 8.
9. 10.
11. 12.
A
A A
A
A
A
B B
B B
B B
18 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.3 Direct Routes – Answers to Exercises
1. 6
2. 15
3. 10
4. 56
5. 70
6. 7
7. 35
8. 126
9. 20
10. 84
11. 21
12. 28
1.4 Barricades ∙∙∙ 21
2006 Vasta & Fisher
1.4 Barricades – Exercises
How many direct routes are there from A to B without crossing any barricades?
1. 2.
3. 4.
5. 6.
A
B
A
B
A
B
A
B
A
B
A
B
22 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
7. 8.
9. 10.
11. 12.
A
B
A
B
A
B
A
B
A
B
A
B
1.4 Barricades ∙∙∙ 23
2006 Vasta & Fisher
13. 14.
A
B
A
B
24 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.4 Barricades – Answers to Exercises
1. 6
2. 8
3. 4
4. 6
5. 14
6. 17
7. 10
8. 16
9. 26
10. 23
11. 20
12. 19
13. 17
14. 18
1.5 Coins & Children ∙∙∙ 27
2006 Vasta & Fisher
1.5 Coins & Children – Exercises
1. Flip a coin 4 times. How many different ways can the outcome have exactly 2 heads?
2. Flip a coin 5 times. How many different ways can the outcome have exactly 3 heads?
3. Flip a coin 6 times. How many different ways can the outcome have exactly 6 heads?
4. Flip a coin 7 times. How many different ways can the outcome have exactly 4 heads?
5. Flip a coin 8 times. How many different ways can the outcome have exactly 6 heads?
6. Flip a coin 9 times. How many different ways can the outcome have exactly 7 heads?
7. In how many ways can a family with 4 children have exactly 3 girls?
8. In how many ways can a family with 5 children have exactly 0 girls?
9. In how many ways can a family with 6 children have exactly 3 girls?
10. In how many ways can a family with 7 children have exactly 2 girls?
11. In how many ways can a family with 8 children have exactly 5 girls?
12. In how many ways can a family with 9 children have exactly 4 girls?
28 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1.5 Coins & Children – Answers to Exercises
1. 6
2. 10
3. 1
4. 35
5. 28
6. 36
7. 4
8. 1
9. 20
10. 21
11. 56
12. 126
Chapter 1 Review ∙∙∙ 29
2006 Vasta & Fisher
Chapter 1 – Problem Solving – Review Exercises
1. Find the next term and classify the sequence as arithmetic, geometric, Fibonacci-
like, or none of these.
4, 8, 12, 16, 20,
2. Construct Pascal’s Triangle up to row 9.
3. How many direct routes are there from A to B?
4. How many direct routes are there from A to B without crossing the barricade?
5. Flip a coin 6 times. How many different ways can the outcome have exactly 4
heads?
6. In how many ways can a family with 5 children have exactly 2 girls?
B
A
A
B
30 ∙∙∙ Chapter 1 Problem Solving
2006 Vasta & Fisher
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1
Chapter 1 – Problem Solving – Review Answers
1. 24, arithmetic
2.
3. 35
4. 26
5. 15
6. 10
2.1 Venn Diagrams ∙∙∙ 35
2006 Vasta & Fisher
2.1 Venn Diagrams – Exercises
Draw a Venn diagram for each of the following relationships.
1. children and senior citizens
2. ladybugs and insects
3. comedy movies and romantic movies
4. one-story houses and red houses
5. reptiles and mammals
6. people, guitarists, and musicians
7. people under 20, people 20 and over, and students
8. deciduous trees, evergreen trees, and pine trees
9. houses, one-story houses, and two-story houses
10. teachers, mothers, runners
11. married men, people, and firefighters
12. dairy products, ice cream, and food
13. mothers, grandmothers, and fathers
14. print books, e-books, and fiction books
15. Cuesta students, students taking a math class, students taking an English class
16. men, people over 40, men over 65
17. females, people under 6, 4-year-old boys
18. males under 30, people under 12, people over 20
36 ∙∙∙ Chapter 2 Set Theory
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2.1 Venn Diagrams – Answers to Exercises
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12.
13. 14. 15. 16.
17. 18.
U = insects
lady bugs
U = people
musicians
guitar-
ists
reptiles mammals
U = living creatures
one-
story red
U = houses
comedy romantic
U = movies
children seniors
U = people
U = people
people
under 20
students
people
20 & over
U = students
Cuesta
math
English
1-story 2-story
U = houses
men over
40
U = people
men over 65
females under 6
U = people
4-yr old boys
U = trees
deciduous
pine trees
evergreen
married
men fire
fighters
U = people U = people
teachers
mothers
runners
U = food
dairy
products ice
cream
U = parents
fathers
grand
mothers
mothers
U = books
books
fiction
e-books
males
under
30
under
12
U = people
over
20
2.2 Set Theory ∙∙∙ 41
2006 Vasta & Fisher
2.2 Set Theory – Exercises
Find the following.
1. {1, 2, 3, 4} {4, 5}
2. {1, 2, 3, 4} {4, 5}
3. {a, b, c} {x, y, z}
4. {a, b, c} {x, y, z}
5. {red, blue, green, yellow} {blue, green}
6. {red, blue, green, yellow} {blue, green}
7. {2, 3, 5, 7} Ø
8. {2, 3, 5, 7} Ø
9. | {2, 3, 5, 7} |
10. | Ø |
Let U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 2, 3, 4, 5}
B = {1, 3, 5, 7}
Find the following.
11. | A |
12. | B |
13. A
14. B
15. A B
16. A B
17. A B
18. A B
19. A B
20. A B
21. | U |
22. U
42 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
Let U = {Bob, Greg, Oliver, Pat, Peter}
A = {Bob, Pat}
B = {Greg, Peter, Bob}
Find the following.
23. | A |
24. | B |
25. A B
26. A B
27. (A B)
28. (A B)
29. U A
30. | U |
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 3, 5}
B = {2, 4, 6}
C = {4, 5, 6, 7}
Find the following.
31. (A B) C
32. A (B C)
33. (A B) C
34. A (B C)
35. (A B) C
36. A (B C)
37. (A B) C
38. (C B) A
39. (A B) (A C)
40. (B C) (A B)
41. | A |
42. | C |
2.2 Set Theory ∙∙∙ 43
2006 Vasta & Fisher
Let U = {a, b, c, d, e, f, g, h}
A = {a, c, e}
B = {b, e, g}
C = {c, d}
Find the following.
43. (A C ) B
44. A (C B)
45. (A C) B
46. (A C) B
47. (A B) (B C)
48. (A B) (B C)
49. A A
50. B B
51. | B C |
52. | A B |
Let A be a subset of a universal set U. Simplify the following.
53. A Ø
54. A Ø
55. A A
56. A A
57. A U
58. A U
59. (A)
60. A A
61. A A
62. U
63.
64. | |
44 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
Definition The difference between A and B is the set that contains the elements in A
but not in B. It is denoted by A – B.
Definition The symmetric difference of A and B is the set that contains the elements
in A or B but not in both. It is denoted by A + B.
Let U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {2, 3, 5, 7}
B = {1, 2, 5, 6}
C = {1, 2, 3, 4, 5}
Find the following.
65. A B
66. A B
67. A – B
68. B – A
69. A + B
70. (A – C) (B + C)
71. (C – A) (B + C)
72. (C – (A (B + (A C))))
73. B + (C (A – (B C)))
74. ((A + C) (B – C)) A
Let A be a subset of a universal set U. Simplify the following.
75. A – Ø
76. Ø – A
77. A + Ø
78. A – A
79. A + A
80. A – U
81. U – A
82. A + U
83. A – A
84. A – A
85. A + A
2.2 Set Theory ∙∙∙ 45
2006 Vasta & Fisher
2.2 Set Theory – Answers to Exercises
1. {1, 2, 3, 4, 5} 2. {4}
3. {a, b, c, x, y, z}
4. Ø
5. {red, blue, green, yellow}
6. {blue, green}
7. {2, 3, 5, 7}
8. Ø
9. 4
10. 0
11. 5
12. 4
13. {6, 7}
14. {2, 4, 6}
15. {1, 3, 5}
16. {1, 2, 3, 4, 5, 7}
17. {1, 3, 5, 6, 7}
18. {7}
19. {1, 2, 3, 4, 5, 6}
20. {2, 4}
21. 7
22. Ø
23. 3
24. 2
25. {Bob, Greg, Oliver, Peter}
26. {Greg, Peter}
27. {Oliver}
28. {Greg, Oliver, Pat, Peter}
29. A
30. 0
31. {4, 5, 6}
32. {1, 3, 4, 5, 6}
33. C
34. {5}
35. Ø
36. {1, 2, 3, 4, 5, 6, 7}
37. {8, 9}
38. {4, 6, 7, 8, 9}
39. {5, 7, 8, 9}
40. {1, 2, 3, 4, 6, 7, 8, 9}
41. 3
42. 5
46 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
43. {a, b, e, g}
44. {a, e}
45. {a, c, d}
46. {a, c, d, f, h}
47. {f, h}
48. U
49. Ø
50. U
51. 0
52. 5
53. A
54. Ø
55. A
56. A
57. U
58. A
59. A
60. U
61. Ø
62. Ø
63. U
64. 0
65. {1, 2, 3, 5, 6, 7}
66. {2, 5}
67. {3, 7}
68. {1, 6}
69. {1, 3, 6, 7}
70. {3, 4, 6, 7}
71. {1}
72. {1, 2, 3, 5, 6, 7, 8}
73. {3, 4, 7, 8}
74. {1, 4, 6}
75. A
76. Ø
77. A
78. Ø
79. Ø
80. Ø
81. A
82. A
83. A
84. A
85. U
2.4 Shading Venn Diagrams ∙∙∙ 61
2006 Vasta & Fisher
2.4 Shading Venn Diagrams – Exercises
Draw a Venn diagram for each of the following sets.
1. A B
2. A B
3. A B
4. A B
5. (A B)
6. (A B)
7. A B
8. A B
9. (A B) C
10. A (B C)
11. (A B) C
12. (A B) (A C)
13. (A B) (A C)
14. (A B) C
15. A (B C)
16. (A B) C
17. A (B C)
18. (A B) C
19. A (B C )
20. (A (B C))
21. (A – B) (B + C)
22. (A + B) – (B C)
23. ((A B) – (C A)) + A
24. (A B) (C D)
25. (A C) (D B)
26. ((A B) C) D
62 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
2.4 Shading Venn Diagrams – Answers to Exercises
1. 2. 3.
4. 5. 6.
7. 8.
9. 10. 11.
12. 13. 14.
2.4 Shading Venn Diagrams ∙∙∙ 63
2006 Vasta & Fisher
15. 16. 17.
18. 19. 20.
21. 22. 23.
24. 25. 26.
2.5 Equal Sets ∙∙∙ 67
2006 Vasta & Fisher
2.5 Equal Sets – Exercises
Are the sets equal?
1. A B (A B)
2. A B (A B)
3. A (B A) B (A B)
4. A (A B) B (A B)
5. A (B C) (A B) (A C)
6. A (B C) (A B) (B C)
7. (A B) C A (B C)
8. (A B) C A (B C )
9. A – B A B
10. (A – B) A B
11. A + B (A – B) (B – A)
12. (A + B) (A B) (A B)
68 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
2.5 Equal Sets – Answers to Exercises
1. No
2. Yes
3. Yes
4. No
5. Yes
6. No
7. No
8. Yes
9. No
10. Yes
11. Yes
12. Yes
2.6 Problem Solving with Sets ∙∙∙ 73
2006 Vasta & Fisher
2.6 Problem Solving with Sets – Exercises
In a survey of 20 people, it is found that 7 own bikes, 10 own cars, and 3 own both.
1. How many of the people surveyed own neither a bike nor a car?
2. How many own a bike but not a car?
3. How many own a bike or a car?
Thirty people fill out a questionnaire at a pet store. The results are that 18 respondents
own cats, 11 own dogs, and 7 own both.
4. How many respondents do not own a cat?
5. How many own neither a cat nor a dog?
6. How many own a cat or a dog but not both?
Of the 25 people who participate in a certain survey, 12 own laptops, 21 own cell phones,
and 9 own both.
7. How many participants owned a cell phone but not a laptop?
8. How many did not own a laptop and a cell phone?
9. How many owned a cell phone or a laptop?
The school cafeteria surveyed 20 students about which fruits they liked. Thirteen of the
students liked apples, 12 liked oranges, and 5 liked both.
10. How many students liked neither apples nor oranges?
11. How many liked both or neither?
12. How many liked apples but not oranges?
In a survey of 100 mathematicians, it is found that 60 brush their teeth daily, 37 floss
daily, and 13 do both.
13. How many mathematicians do not brush and floss?
14. How many do not floss?
15. How many do both or neither?
Of the 40 students who participate in a certain survey, 29 use pens, 18 use pencils, and 8
use both.
16. How many students used a pen or a pencil but not both?
17. How many used neither a pencil nor a pen?
18. How many used a pencil but not a pen?
74 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
Joe questions 35 students to determine whether they have a cat, a dog, or a bird. He gets
the following results.
21 have a cat
20 have a dog
18 have a bird
12 have a cat and a dog
11 have a cat and a bird
10 have a bird and a dog
8 have all three
19. Draw a Venn diagram representing this information.
20. How many students have only a cat?
21. How many do not have a dog?
22. How many have a cat or a bird?
23. How many have a cat and a bird?
24. How many have only one of the three?
25. How many have a cat and a dog, but not a bird?
26. How many have a cat or a dog, but not a bird?
27. How many do not have any of the three?
In a survey of 100 students, the following information was found.
40 students like history
24 like science
30 like math
8 like history and science
9 like science and math
10 like history and math
3 like all three classes
28. Draw a Venn diagram representing this information.
29. How many students did not like any of these classes?
30. How many liked only history?
31. How many did not like science?
32. How many liked history and science, but not math?
33. How many liked history or science, but not math?
34. How many liked at least two of these classes?
35. How many liked math or history?
36. How many liked math and history?
2.6 Problem Solving with Sets ∙∙∙ 75
2006 Vasta & Fisher
In a survey of 45 people, the following information was found.
33 students like chocolate
18 like peanut butter
7 like lima beans
11 like chocolate and peanut butter
5 like peanut butter and lima beans
6 like chocolate and lima beans
4 like all three foods
37. How many people did not like any of these foods?
38. How many liked only lima beans?
39. How many liked chocolate but not peanut butter?
40. How many liked chocolate or lima beans, but not peanut butter?
41. How many liked chocolate and lima beans, but not peanut butter?
42. How many liked only one of the three?
43. How many did not like chocolate?
44. How many liked chocolate and peanut butter?
Sixty male math nerds were interviewed, and the following information was found.
44 male math nerds like algebra
35 like calculators
6 like girls
2 like calculators and girls
22 like calculators and algebra
5 like algebra and girls
2 like all three things
45. How many male math nerds only liked girls?
46. How many did not like algebra?
47. How many liked at least two of these things?
48. How many did not like any of these things?
49. How many liked calculators but not girls?
50. How many liked algebra and girls, but not calculators?
51. How many liked algebra or girls, but not calculators?
52. How many did not like girls?
76 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
In a survey of 80 T.V. viewers, the following information was found.
26 viewers like ABC
38 like CBS
31 like NBC
29 like FOX
12 like ABC and CBS
11 like ABC and NBC
7 like ABC and FOX
16 like CBS and NBC
12 like CBS and FOX
10 like NBC and FOX
6 like ABC, CBS, and NBC
3 like ABC, CBS, and FOX
5 like ABC, NBC, and FOX
4 like CBS, NBC, and FOX
1 likes all four networks
53. How many liked only FOX?
54. How many did not like ABC?
55. How many did not like any of the four networks?
56. How many liked only one of the four networks?
57. How many liked ABC and CBS, but not FOX?
58. How many liked ABC or CBS, but not FOX?
2.6 Problem Solving with Sets ∙∙∙ 77
2006 Vasta & Fisher
2.6 Problem Solving with Sets – Answers to Exercises
1. 6 2. 4
3. 14
4. 12
5. 8
6. 15
7. 12
8. 16
9. 24
10. 0
11. 5
12. 8
13. 87
14. 63
15. 29
16. 31
17. 1
18. 10
19.
20. 6
21. 15
22. 28
23. 11
24. 17
25. 4
26. 16
27. 1
28.
78 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
29. 30
30. 25
31. 76
32. 5
33. 40
34. 21
35. 60
36. 10
37. 5
38. 0
39. 22
40. 22
41. 2
42. 26
43. 12
44. 11
45. 1
46. 16
47. 25
48. 2
49. 33
50. 3
51. 23
52. 54
53. 11
54. 54
55. 7
56. 38
57. 9
58. 36
Chapter 2 Review ∙∙∙ 79
2006 Vasta & Fisher
Chapter 2 – Set Theory – Review Exercises
1. Draw a Venn diagram showing the relationship among kittens and dogs.
Find the following.
2. {a, e, i, o, u} {a, b, c, d, e}
3. {a, e, i, o, u} {a, b, c, d, e}
4. | {a, e, i, o, u} |
Let U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 3, 4}
B = {1, 3, 5, 7}
C = {4, 5, 6, 7}
Find the following.
5. (A C) B
6. | A B |
7. Let A be a subset of a universal set U. Simplify the following.
(A Ø) A
8. Draw a Venn diagram and shade the region representing A B.
9. Draw a Venn diagram and shade the region representing (A B) C .
10. Are the following sets equal? (A B) A B
11. Are the following sets equal? (A B) C (A C) (B C)
For a biology report, Jimmy questions 14 of his relatives. He finds that 8 can curl their
tongue, 5 can wiggle their ears, and 3 can do both.
12. How many of the people surveyed can curl their tongue or wiggle their ears?
13. How many can’t curl their tongue or wiggle their ears?
14. How many can wiggle their ears but not curl their tongue?
80 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
A small company questions 50 of its employees to find out which methods of
transportation they have used to commute to work during the last month.
38 have driven to work
24 have bicycled to work
15 have walked to work
16 have driven to work and bicycled to work
10 have driven to work and walked to work
7 have bicycled to work and walked to work
4 have commuted to work all three ways
15. Draw a Venn diagram representing this information.
16. How many employees have not commuted to work any of the three ways?
17. How many have not biked to work?
18. How many have only walked to work?
19. How many have walked to work and biked to work?
20. How many have commuted to work only one of the three ways?
21. How many have driven or walked to work?
22. How many have driven to work or biked to work, but not walked to work?
23. How many have walked to work and biked to work, but not driven to work?
Chapter 2 Review ∙∙∙ 81
2006 Vasta & Fisher
Chapter 2 – Set Theory – Review Answers
1.
2. {a, b, c, d, e, i, o, u}
3. {a, e}
4. 5
5. {8}
6. 6
7. Ø
8.
9.
10. Yes
11. No
12. 10
13. 4
14. 2
kittens dogs
U = animals
82 ∙∙∙ Chapter 2 Set Theory
2006 Vasta & Fisher
15.
16. 2
17. 26
18. 2
19. 7
20. 23
21. 43
22. 33
23. 3