1.1 sequences exercises - wordpress.com6 ∙∙∙ chapter 1 problem solving & set theory 2006...

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 & Set Theory

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



10. 192, geometric

11. –2, arithmetic


13. 7, arithmetic

14. –7, none of these

15. 1/81, geometric






1.2 Pascal’s Triangle ∙∙∙ 11

2006 Vasta & Fisher

1.2 Pascal's Triangle – Exercises

Construct Pascal's Triangle up to row 9.


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


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


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


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


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?






7. In how many ways can a family with 4 children have exactly 3 girls?







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

1.6 Venn Diagrams ∙∙∙ 33

2006 Vasta & Fisher

1.6 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. males, females, and little girls

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. paperback books, hardback 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


2006 Vasta & Fisher

1.6 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 = people

males

little girls

females

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

paper-

back

fiction

hard-

back

males

under

30

under

12

U = people

over

20

1.7 Set Theory ∙∙∙ 39

2006 Vasta & Fisher

1.7 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


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 |


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. | |


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


2006 Vasta & Fisher

1.7 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


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

1.9 Shading Venn Diagrams ∙∙∙ 59

2006 Vasta & Fisher

1.9 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


2006 Vasta & Fisher

1.9 Shading Venn Diagrams – Answers to Exercises

1. 2. 3.

4. 5. 6.

7. 8.

9. 10. 11.

12. 13. 14.

1.9 Shading Venn Diagrams ∙∙∙ 61

2006 Vasta & Fisher

15. 16. 17.

18. 19. 20.

21. 22. 23.

24. 25. 26.

1.10 Equal Sets ∙∙∙ 65

2006 Vasta & Fisher

1.10 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)


2006 Vasta & Fisher

1.10 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

1.11 Problem Solving with Sets ∙∙∙ 71

2006 Vasta & Fisher

1.11 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?


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


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?


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?


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?


2006 Vasta & Fisher

1.11 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.


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 1 Review ∙∙∙ 77

2006 Vasta & Fisher

Chapter 1 – Problem Solving & Set Theory – 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?


7. Draw a Venn diagram showing the relationship among kittens and dogs.

8. {a, e, i, o, u} {a, b, c, d, e}

9. {a, e, i, o, u} {a, b, c, d, e}

10. | {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.

11. (A C) B

12. | A B |

B

A

A

B


2006 Vasta & Fisher

13. Let A be a subset of a universal set U. Simplify the following.

(A Ø) A

14. Draw a Venn diagram and shade the region representing A B.

15. Draw a Venn diagram and shade the region representing (A B) C .

16. Are the following sets equal? (A B) A B

17. 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.

18. How many of the people surveyed can curl their tongue or wiggle their ears?

19. How many can’t curl their tongue or wiggle their ears?

20. How many can wiggle their ears but not curl their tongue?

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


22. How many employees have not commuted to work any of the three ways?

23. How many have not biked to work?

24. How many have only walked to work?

25. How many have walked to work and biked to work?

26. How many have commuted to work only one of the three ways?

27. How many have driven or walked to work?

28. How many have driven to work or biked to work, but not walked to work?

29. How many have walked to work and biked to work, but not driven to work?

Chapter 1 Review ∙∙∙ 79

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 & Set Theory – Review Answers

1. 24, arithmetic

2.

3. 35

4. 26

5. 15

6. 10

7.

8. {a, b, c, d, e, i, o, u}

9. {a, e}

10. 5

11. {8}

12. 6

kittens dogs

U = animals


2006 Vasta & Fisher

13. Ø

14.

15.

16. Yes

17. No

18. 10

19. 4

20. 2

21.

22. 2

23. 26

24. 2

25. 7

26. 23

27. 43

28. 33

29. 3

1.1 sequences exercises - wordpress.com6 ∙∙∙ chapter 1 problem solving & set theory 2006...

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