A. M and D B. M and V C. M and F D. V and F 6. Which Venn diagram correctly represents the situation described? Rahim described the set as follows:

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1 Multiple Choice 1. What is the universal set? A. a set with an infinite number of elements B. a set of all the elements under consideration for a particular context C. a set with a countable number of elements D. a set that contains every possible element 2. Which pair of sets represents disjoint sets? A. N, the set of natural numbers, and I, the set of integers B. T, the set of all triangles, and C, the set of all circles C. N, the set of natural numbers, and P, the set of positive integers D. none of the above 3. Which pair of sets represents equal sets? A. N, the set of natural numbers, and I, the set of integers B. T, the set of all triangles, and C, the set of all circles C. N, the set of natural numbers, and P, the set of positive integers D. none of the above 4. Which pair of sets represents one set being a subset of another but is not equal? A. N, the set of natural numbers, and I, the set of integers B. T, the set of all triangles, and C, the set of all circles C. N, the set of natural numbers, and P, the set of positive integers D. none of the above 5. Rahim described the set as follows: M = {all of the foods he eats} D = {his favourite desserts} V = {his favourite vegetables} F = {his favourite fruits} Which are the disjoint sets? A. M and D B. M and V C. M and F D. V and F 6. Which Venn diagram correctly represents the situation described? Rahim described the set as follows: M = {all of the foods he eats} D = {his favourite desserts} V = {his favourite vegetables} F = {his favourite fruits} Assume Rahim likes some fruit for dessert. A. B. C. D. 7. Given the following situation: June

2 the universal set U = {positive integers less than 20} X = {4, 5, 6, 7, 8} P = {prime numbers of U} O = {odd numbers of U} Which statement describes O? A. the set of even numbers of U B. the set of odd numbers of U C. the set of odd, prime numbers of U D. the set of even, prime numbers of U 8. There are 28 students in Mr. Connelly s Grade 12 mathematics class. The number of students in the yearbook club and the number of students on student council are shown in the Venn diagram. Use the diagram to answer the following questions. (#8 to 12) How many students are in both the yearbook club and on the student council? A. 2 B. 5 C. 1 D How many students are in the yearbook club but not on student council? A. 2 B. 5 C. 1 D How many students are in at least one of the yearbook club or on student council? A. 2 B. 5 C. 8 D How many students are on the student council but not in the yearbook club? A. 2 B. 5 C. 1 D How many students are neither in the yearbook club nor on student council? A. 2 B. 5 C. 1 D Consider the following Venn diagram of herbivores and carnivores: (#13 to 16) June

3 Determine H C. A. {moose, rabbit, deer, squirrel} B. {bear, raccoon, badger} C. {cougar, wolf} D. {moose, rabbit, deer, squirrel, bear, raccoon, badger, cougar, wolf} 14. Determine n(h C). A. 2 B. 9 C. 4 D Determine H C. A. {moose, rabbit, deer, squirrel} B. {bear, raccoon, badger} C. {cougar, wolf} D. {moose, rabbit, deer, squirrel, bear, raccoon, badger, cougar, wolf} 16. Determine n(h C). A. 2 B. 9 C. 4 D Consider the following two sets: C = { 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10} B = { 9, 6, 3, 0, 3, 6, 9, 12} Determine n(c B). A. 3 B. 8 C. 11 D Consider the following two sets: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B = { 9, 6, 3, 0, 3, 6, 9, 12} Determine n(a B). A. 8 B. 11 C. 16 D The three circles in the Venn diagram (P, Q, and R ) contain the same number of elements. Which set of values is true for p, q, and r? June

4 A. p = 11, q = 11, r = 5 B. p = 7, q = 8, r = 2 C. p = 7, q = 6, r = 1 D. p = 14, q = 13, r = A summer camp offers canoeing, rock climbing, and archery. The following Venn diagram shows the types of activities the campers like. (#20 to 23) Use the diagram to determine n((r C) \ A). A. 64 B. 48 C. 37 D Use the diagram to determine n((r A) \ C). A. 14 B. 5 C. 26 D Use the diagram to determine n((a C) (R A)). A. 16 B. 27 C. 5 D Use the diagram to determine n((a \ C) \ R). A. 8 B. 22 C. 6 D Some table games use a board, dice, or cards, or a combination these. The following Venn diagram shows the number of games that use these tools. (#24 to 28) June

5 Use the diagram to determine n(u). A. 100 B. 104 C. 97 D Use the diagram to determine n(c) n(c B). A. 39 B. 51 C. 25 D Use the diagram to determine the number of games that use exactly two of these tools. A. 13 B. 51 C. 25 D Use the diagram to determine n(b \ D). A. 44 B. 33 C. 67 D Use the diagram to determine n(d B). A. 29 B. 25 C. 69 D. 75 Short Answer 1. Tania recorded the 16 possible sums that can occur when you roll two four-sided dice. S = {all possible sums} L = {all sums less than 4} G = {all sums greater than 4} F = {all sums equal to 4} What is n(l)? 2. Tania recorded the 16 possible sums that can occur when you roll two four-sided dice. S = {all possible sums} June

6 L = {all sums less than 4} G = {all sums greater than 4} F = {all sums equal to 4} a.) b.) What is n(g)? What is n(f)? c.) What is the number of sums less than or greater than 4? d.) Describe the relationship between the sets L and G. e.) List the subsets using set notation. f.) Describe the relationship between events L and G. 3. What is the set notation for the set of all natural numbers greater than 1 and less than or equal to 50? 4. Carlos surveyed 50 students about their favourite subjects in school. He recorded his results. Favourite Subject Number of Students mathematics 18 science 15 neither mathematics nor science 20 a.) b.) Determine how many students like mathematics and science. Determine how many students like only mathematics or only science. 5. Mrs. Lam s physics class is visiting the local amusement park. She has 32 students. Of these students, 20 plan to ride the roller coaster and 15 plan to ride the vertical drop. There are 8 students who do not plan to ride either attraction. Determine how many students plan to ride both the roller coaster and the vertical drop. 6. Mr. McSherry s physics class is visiting the local amusement park. He has 33 students. Of these students, 21 plan to ride the roller coaster and 12 plan to ride the vertical drop. There are 7 students who do not plan to ride either attraction. Determine how many students plan to ride only the roller coaster. 7. If you draw a card at random from a standard deck of cards, you will draw a card that is either red (R) or black (B). The card will also either be a number card (N) or a face card (F). Determine n(r F). 8. Games that use a board include chess, Clue, checkers, Go, Scrabble, and Monopoly. Games that use cards include Hearts, Monopoly, Snap, and Clue. Draw a Venn diagram to represent these two sets of games. 9. Grade 12 students were surveyed about their extra curricular activities. 58% belonged to a sports team (S) June

7 63% belonged to a band or choir (B) 47% belonged to a school club (C) 24% belonged to a sports team and a band or choir 21% belonged to a sports team and a school club 36% belonged to a band or choir and a school club 19% engaged in all three activities a.) b.) c.) d.) e.) What percent of students only belong to a band or choir? Write your answer in set notation. What percent of students belong to a sports team or a school club but not a band or choir? Write your answer in set notation. What percent of students belong to both a sports team and a band or choir but not a school club? Write your answer in set notation. What percent of students are involved in only one of these activities? Write your answer in set notation. What percent of students are involved in only two of these activities? Write your answer in set notation. 10. The city surveyed 3000 people about how they travel to work took public transit (P) 1494 drove (D) 818 cycled (C) 731 took public transit and drove only 298 took public transit and cycled only 27 drove and cycled only 164 used all three modes of transportation a.) b.) c.) d.) e.) f.) g.) Create a Venn diagram to show your answer. How many people travel to work some other way? How many people use public transit only? Use a Venn diagram to show your answer. How many people use two modes of transportation? Use a Venn diagram to show your answer. How many people did not cycle or use public transit? Write your answer using set notation. How many people drove or used public transit but did not cycle? Write your answer using set notation. How many people used only one mode of transport? Write your answer using set notation. h.) How many people only cycle? Write your answer using set notation. Problem i.) How many people cycle but do not drive? Write your answer using set notation June

8 1. a) Indicate the multiples of 2 and 3, from 1 to 100, using set notation. List any subsets. b) Represent the sets and subsets in a Venn diagram. 2. Paul asked 30 people who saw a movie based on a popular book if they liked the book or the movie. 3 people did not like the movie or the book. 15 people liked the movie. 22 people liked the book. Determine how many people liked both the movie and the book, how many liked only the movie, and how many liked only the book. 3. A sports radio station polled 500 listeners about their favourite sports. 245 people listed hockey. 213 people listed basketball. 84 people did not list hockey or basketball. Determine how many people listed both hockey and basketball, how many listed hockey but not basketball, and how many listed basketball but not hockey. 4. Dorothy asked 50 people outside a bookstore if they preferred physical books or electronic readers. 2 people said they did not read books. 20 people liked both physical books and electronic readers. 13 people liked only physical books. Determine how many people liked only electronic readers. 5. A game store polled 150 customers about whether they preferred to play strategy games or games of chance. 75 people liked to play both. 20 people did not like games. 31 people liked only strategy games. Determine how many people liked only games of chance. 6. Noreen asked 65 people at a gym if they liked cycling or running. 7 people did not like either activity. 24 people liked cycling. 40 people liked running. Determine how many people liked cycling and running. 7. A pet store polled 500 customers about whether they preferred cats or dogs. 273 people liked cats. 264 people liked dogs. 9 people did not like either cats or dogs. Determine how many people liked cats and dogs. 8. A camping store surveyed 150 people about the national parks they had visited. 91 people had been to Banff National Park in Alberta. 77 people had been to Glacier National Park in British Columbia. 36 people had not been to either park. Determine how many people had been to both parks. Draw a Venn diagram to show your solution. 9. A camping store surveyed 450 people about whether they liked to stay in a cabin, a tent, or both. 316 people liked a tent. 193 people liked a cabin. 63 people did not like either option. Determine how many people liked staying in a tent and a cabin. Draw a Venn diagram to show your solution. 10. A total of 83 teens attended a performing arts camp to train in at least one of three activities: dance, acting, or singing. June

9 47 took dance, 42 took acting, and 54 took singing. 3 took dance and acting only. 16 took acting and singing only. 19 took dance and singing only. How many teens trained in all three performing arts? 11. A total of 83 teens attended a performing arts camp to train in at least one of three activities: dance, acting, or singing. 50 took dance, 50 took acting, and 44 took singing. 15 took dance and acting only. 12 took dance and singing only. 69 took acting or singing or both. 9 took all three performing arts. How many teens trained in only one performing art? Use a Venn diagram in your answer. 12. The card game Uno has cards divided into 4 colours: red, blue, green, and yellow. For each colour, there are 19 number cards and 6 action cards. There are also 8 special black action cards. Determine the following amounts. a) n(d), the total number of cards in the deck b) n(a), the total number of action cards in the deck c) n(g), the total number of green cards in the deck d) n(n), the total number of number cards in the deck e) n(a G) f ) n(g N) Math 3201 Chapter 1 Final Review Answer Section MULTIPLE CHOICE 1. B 8. A 15. B 22. A 2. B 9. B 16. D 23. C 3. C 10. C 17. A 24. B 4. A 11. C 18. C 25. D 5. D 12. D 19. C 26. C 6. D 13. D 20. B 27. D 7. A 14. B 21. D 28. A June

10 SHORT ANSWER 1. n(l) = 3 2. a.) n(g) = 10 b.) n(f) = 3 c.) n(l or G) = 13 d.) Sets L and G are disjoint. e.) L S, G S, F S f.) Events L and G are mutually exclusive. 3. A = {x 1 < x 50, x N} 4. a.) Three students like mathematics and science. b.) Fifteen students like only mathematics and 12 students like only science. 5. There are 11 students who plan to ride both the roller coaster and the vertical drop. 6. There are 14 students who plan to ride only the roller coaster DIF: Grade 12 REF 9. a.) n(b \ S \ C) = 22% b.) n((s C) \ B) = 43% c.) n((b S) \ C) = 5% d.) n(b \ S \ C) + n(s \ B \ C) + n(c \ S \ B) = 63% e.) n((s B) \ C) + n((c S) \ B) + n((b C) \ S) = 24% 10. a.) b.) 94 people travel to work some other way. c.) 785 people use public transit only. d.) 1056 people use two modes of transportation. e.) n(c P) = 666 f.) n((d P) \ C) = 2088 g.) n(d \ P \ C) + n(p \ D \ C) + n(c \ P \ D) = 1686 h.) n(c \ P \ D) = 329 i.) n(c \ D) = 627 June

11 PROBLEM 1. ANS: a) S = {1, 2, 3,, 98, 99, 100} S = {x 1 x 100, x N} D = {2, 4, 6,, 96, 98, 100} D = {d d = 2x, 1 x 50, x N} D S T = {3, 6, 9,, 93, 96, 99} T = {t t = 3x, 1 x 33, x N} T S b) 2. ANS: Let U represent the universal set. Let M represent the set of people who liked the movie. Let B represent the set of people who liked the book. n(m B) = n(u) n(m B) n(m B) = 30 3 n(m B) = 27 n(m) + n(b) = n(m) + n(b) = 37 n(m B) = n(m B) = people liked both the book and the movie. n(m only) = n(m only) = 5 5 people liked the movie only. n(b only) = n(b only) = people liked the book only. 3. ANS: Let U represent the universal set. Let H represent the set of people who listed hockey. Let B represent the set of people who listed basketball. n(h B) = n(u) n(h B) n(h B) = n(h B) = 416 n(h) + n(b) = n(h) + n(b) = 458 n(h B) = n(h B) = people listed both hockey and basketball. n(h only) = n(h only) = people listed hockey but not basketball. n(b only) = n(b only) = people listed basketball but not hockey. June

12 4. ANS: Let U represent the universal set. Let R represent the set of people who liked physical books. Let E represent the set of people who liked electronic readers. n(e only) = n(u) n(e R) n(r only) n(e R) n(e \ R) = n(e \ R) = people like electronic readers only. 5. ANS: Let U represent the universal set. Let S represent the set of people who liked strategy games. Let C represent the set of people who liked games of chance. n(c \ S) = n(u) n(c S) n(s only) n(c S) n(c \ S) = n(c \ S) = people like games of chance only. 6. ANS: Let U represent the universal set. Let C represent the set of people who liked cycling. Let R represent the set of people who liked running. n(c R) = n(u) n(c R) n(c R) = 65 7 n(c R) = 58 n(c R) = n(c) + n(r) n(c R) n(c R) = n(c R) = 6 6 people like cycling and running. 7. ANS: Let U represent the universal set. Let C represent the set of people who liked cats. Let D represent the set of people who liked running. n(c D) = n(u) n(c D) n(c D) = n(c D) = 491 n(c D) = n(c) + n(d) n(c D) n(c D) = n(c D) = people like cats and dogs. 8. ANS: Let U represent the universal set. Let B represent the set of people who visited Banff National Park. Let G represent the set of people who visited Glacier National Park. n(b G) = n(u) n(b G) n(b G) = n(b G) = 114 n(b G) = n(b) + n(g) n(b G) n(b G) = n(b G) = 54 June

13 9. ANS: Let U represent the universal set. Let T represent the set of people who like tents. Let C represent the set of people who like cabins. n(t C) = n(u) n(t C) n(t C) = n(t C) = 387 n(t C) = n(t) + n(c) n(t C) n(t C) = n(t C) = ANS: Let x represent the number of teens trained in all three performing arts. Using the principle of inclusion and exclusion for three sets: 11 teens are studying all three performing arts. 11. ANS: Calculate dance only. n(d \ A \ S): = 14 Record the known numbers on a Venn diagram. Let x represent the number of teens trained in acting and singing but not dance. Using the principle of inclusion and exclusion for three sets: n(a \ D \ S): = 10 n(s \ A \ D): = 7 14 students took dance only, 10 students took acting only, and 7 students took singing only. June

14 12. ANS: a) n(d), the total number of cards in the deck: There are 4 colours with 19 number cards and 6 action cards, plus 8 black cards. 4(19 + 6) + 8 = 108 There are 108 cards in the deck. b) n(a), the total number of action cards in the deck: There are 4 colours with 6 action cards, plus 8 black action cards. 4(6) + 8 = 32 There are 32 action cards. c) n(g), the total number of green cards in the deck: In green, there are 19 number cards and 6 action cards = 25 There are 25 green cards. d) n(n), the total number of number cards in the deck. There are 4 colours with 19 number cards each. 4(19) = 76 There are 76 number cards. e) n(a G): there are 32 action cards and 25 green cards, but 6 action cards are also green = 51 There are 51 cards that are green or action cards. f) n(g N): there are 25 green cards and 76 number cards but only 19 green number cards. June