Question Bank | Unit 12 - Sets and Venn Diagrams

1. Given that $P = {a, b, c, d}$ , $Q = {b, c, d, e, f}$ and $R = {a, c, f, g}$ , mark the members of these sets clearly on the Venn diagram given. Using your diagram, or otherwise, list the members of $(P \cup Q) \cap R$ .

2. (a) $ℰ = {1, 3, 5, 7, 9, 11}$ , $X = {1, 5, 9}$ and $Y = {3, 9, 11}$ . List the elements of

(i) ${X′}^{} \cap Y$ ,

(ii) $(X \cap Y {)'}^{}$ .

(b) In the Venn diagram $ℰ$ is the set of all children in a certain chosen group, $A = {children in Youth Club A}$ and $B = {children in Youth Club B}$ . The letters $p$ , $q$ , $x$ and $y$ in the diagram represent the number of children in each subset. Given that $n (ℰ) = 200$ , $n (A) = 75$ and $n (B) = 35$ ,

(i) express $p$ in terms of $x$ ,

(ii) find the smallest possible value of $y$ ,

(iii) find the largest possible value of $x$ ,

(iv) find the value of $q$ if $p = 45$ .

3. Given that $ℰ = {x : x is an integer, 2 \leq x \leq 10}$ , $A = {x : x is a prime number}$ and $B = {x : x is a multiple of 3}$ ,

(a) find $n (A \cup B)$ ,

(b) list the elements of the set ${A′}^{} \cap {B′}^{}$ .

4. In the Venn diagram (a) shade $A \cap C′$ .

In the Venn diagram (b) shade $B \cup (A \cap C′)$ .

5. At an international college, students have to be proficient in at least one of the languages, English, French or German. In a particular group of $33$ students, $2$ are proficient in all three languages, $3$ in English and French only, $4$ in French and German only and $5$ in German and English only. The number of students proficient in English only is $x$ , in French only is $x$ , and in German only is $(x + 1)$ . Illustrate this information on a clearly labelled Venn diagram, showing the number in each separate region. Find $x$ and hence find the total number proficient in English.

6. In the Venn diagram (a), shade $(A \cap B) \cup C$ .

In the Venn diagram (b), shade $(A \cap C) \cup (B \cap C′)$ .

7. Two sets $A$ and $B$ are such that $n (A) = 11$ and $n (B) = 6$ . Given that $n (ℰ) = 15$ , find

(a) the smallest possible value of $n (A \cap B)$ ,

(b) the largest possible value of $n (A \cup B)'$ .

8. A number of travellers were questioned about the transport they used on a particular day. Each of them used one or more of the methods shown in the Venn diagram. Of those questioned, $6$ said that they travelled by bus and train only, $2$ by train and car only and $7$ by bus, train and car. The number $x$ who travelled by bus only was equal to the number who travelled by bus and car only.

(a) Given that $35$ people used buses and $25$ people used trains, find

(i) the value of $x$ ,

(ii) the number who travelled by train only,

(iii) the number who travelled by at least two methods of transport.

(b) Given also that $85$ people were questioned altogether, calculate the number who travelled by car only.

9. $P = {Parallelograms}$ , $R = {Rectangles$ and $H = {Rhombuses}$ . Draw, in the answer space, a clearly labelled Venn diagram to show the relation between the sets $P$ , $R$ and $H$ .

10. $A$ , $B$ and $C$ are three sets and $ℰ = A \cup B \cup C$ . The number of elements in each subset is shown in the Venn diagram.

(a) Find $n (B)$ .

(b) Given that $n (A) = 41$ , find $x$ .

(c) Find $n (A′ \cap B \cap C)$ .

11. $ℰ = {p, q, r, s, t}$ , $A = {r, t}$ , $B = {q, r, s}$ and $C = {q, s}$ .

(a) State the value of $n (B \cup C)$ .

(b) List the members of (i) $A \cap B$ and (ii) $A′ \cap (B \cup C)$ .

12. It is given that $ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9}$ , $A = {2, 4, 6, 8}$ and $B = {3, 6, 9}$ .

(a) Find the value of $n (A \cap B)$ .

(b) List the members of $(A \cup B)'$ .

13. (a) Three sets, $A$ , $B$ and $C$ , satisfy the following conditions: $n (A \cap B) \neq 0$ , $C \subseteq B$ , $A \cap C = \emptyset$ .

Draw and label a Venn diagram to illustrate these three sets.

(b) Two sets, $F$ and $G$ , satisfy the following conditions: $F = {(x, y) : y = 5 x + 7}$ , $G = {(x, y) : y = m x + c}$ , $F \cap G = \emptyset$ . Write down the value of $m$ and a possible value for $c$ .

14. In a comprehensive school all 200 children in the first year study either Physics, or Chemistry, or both Physics and Chemistry. Given that 80% study Physics and 30% study Chemistry, find the number of children who study

(a) both Physics and Chemistry,

(b) Physics only.

15. (a) $ℰ = {2, 3, 4, 6, 8, 9}$ , $A = {prime numbers}$ , $B = {even numbers}$ , $C = {multiples of 3}$ . List the elements of the sets (i) $A \cap B$ , (ii) $(A \cap C′) \cup B′$ .

(b) The sets $P$ , $Q$ and $R$ satisfy the conditions $Q \subset P$ , $n (Q \cap R) = 0$ and $n (P \cap R) = n (R)$ .

Draw one clearly labelled Venn diagram to illustrate the sets $P$ , $Q$ and $R$ .

16. It is given that $ℰ = {a, b, c, d, e, f}$ , $P = {a, c, e}$ and $Q = {b, c, d, e}$ .

(a) Find $n (P \cap Q)$ .

(b) List the members of $(P \cup Q)'$ .

(c) List the members of $P′ \cap Q$ .

17. All the pupils in a class have four choices in their study of languages. They may study French only, or French and German, or French and Latin, or all three of these languages. Draw a single clearly labelled Venn diagram to illustrate this information.

18. $A$ , $B$ and $C$ are three sets and $ℰ = A \cup B \cup C$ . The numbers of elements in some of the subsets are shown in the Venn diagram and $n (ℰ) = 66$ . Find

(a) $n (A \cup B)$ ,

(b) $n (C)$ ,

(c) $n (A \cap B′)$ .

19. (a) In a survey carried out for a television company, the viewing choices of 100 families, on a particular evening, were recorded. 46 families said they had watched "The Syndicate", 62 families said they had watched "Casualty Ward". 13 families said they had not watched either programme. By drawing a Venn diagram, or otherwise, calculate (i) the number of families who had watched both programmes, (ii) the number of families who had watched "The Syndicate" only.

(b) It is given that $ℰ = {x : x is an integer, 1 \leq x \leq 16}$ , $A = {x : x is a perfect square}$ , $B = {x : x is a multiple of 3}$ , $C = {x : x is a prime number}$ ,

(i) List the elements of $A$ .

(ii) Find $n (A \cap B \cap C)$ .

(iii) List the elements of $(A \cup B \cup C)'$ .

20. (a) In the Venn Diagram in the answer space, shade $A \cup (B \cap C)$ .

(b) $P$ and $Q$ are two sets and $P \cup Q = ℰ$ . Given that $n (P \cap Q) = 9$ , $n (P) = 16$ and $n (P') = 12$ , find (i) $n (P \cup Q)$ , (ii) $n (Q')$ .

21. Given that $ℰ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}$ , list the elements of the following sets:

(a) ${factors of 66}$ ,

(b) ${x : \frac{1}{2} x \geq 5 \frac{1}{2}}$ ,

(c) ${y : y^{3} \leq 150}$ .

22. There are 25 children in a class. Of these, 12 are in the School Play and 18 are in the School Choir. It is given that $ℰ = {children in the class}$ , $P = {children in the School Play}$ , and $C = {children in the School Choir}$ .

(a) Find $n (P')$ .

(b) Find the smallest possible value of $n (P \cap C)$ .

(c) Express in set notation ${children who are neither in the School Play nor the School Choir}$ .

23. $ℰ = {x : x is an integer, 1 \leq x \leq 10}$ , $A = {x : x - 1 \geq 3}$ , $B = {x : 8 < 4 x < 30}$ . List the elements of the sets (a) $A$ , (b) $B$ , (c) $(A \cup B)'$ .

24. In a school, some of the subjects that it is possible to take are Mathematics, Additional Mathematics and Physics. The Venn Diagram shows the combinations of these subjects that are possible and the numbers and letters represent the numbers of students in each subject.

(a) Given that the numbers of students taking Physics is 123, calculate the value of $x$ .

(b) Given that one sixth of those taking Mathematics also take Additional Mathematics, calculate the value of $y$ and hence find the total number of students taking Mathematics.

25. (a) $ℰ = {10, 11, 12, 13, 14, 15, 16, 17}$ , $A = {x : x is a multiple of 3}$ , $B = {x : x is a multiple of 5}$ , $C = {x : x is a multiple of 7}$ .

(i) Find $n (A \cup B)$ ,

(ii) List the elements of the set $(A' \cap B′ \cap C')$ .

(b) In the Venn diagram in the answer space, shade $(P \cap Q) \cup R'$ .

26. In a school, 120 boys play cricket. $ℰ$ is the set of all boys who play cricket, $X$ is the set of batsmen, $Y$ is the set of bowlers. The letters $a$ , $b$ , and $c$ in the Venn diagram represent the number of boys in each subset of $X$ and $Y$ . The letter $d$ represents the number of boys who are neither batsmen nor bowlers. Given that $n (ℰ) = 120$ , $n (X) = 80$ and $n (Y) = 48$ , find

(a) the value of $b$ if $d = 0$ ,

(b) the value of $d$ if $b = c$ ,

(c) the largest possible number of boys who are neither batsmen nor bowlers.

27. $A$ , $B$ and $C$ are three sets and $ℰ = A \cup B \cup C$ . The numbers in the diagram represent the number of elements in each subset.

(a) Given that $n (B) = n (C)$ , find the value of $x$ .

(b) Find $n (B \cup C)'$ .

28. Each of the 56 pupils in the Fourth Year of a small school studies at least one of the subjects History, English and Agriculture. Of the 14 pupils who study Agriculture, 4 also study History and English, 3 study neither History nor English and 5 study English but not History. Of the 42 pupils who do not study Agriculture, 6 study both History and English, $x$ study only History and $2 x$ study only English. Copy the given Venn Diagram and on your copy show the number of pupils in each subset. Hence find

(a) the value of $x$ ,

(b) the total number of pupils studying English.

29. $A$ , $B$ and $C$ are three sets and $ℰ = A \cup B \cup C$ . The numbers in the diagram represent the number of elements in each subset.

(a) Given that $n (A) = n (B)$ , find the value of $x$ .

(b) Find $n (A \cup B)'$ .

30. $L$ , $M$ and $N$ are three sets and $ℰ = L \cup M \cup N$ . It is given that $n (ℰ) = 85$ , $n (L \cap M) = 12$ , $n (M \cap N) = 15$ and $n (L) = n (M) = 30$ .

(a) Copy the Venn diagram onto your answer paper and use the given information to write the number of elements in each subset.

(b) Find the value of $n (M')$ .

(c) Find the value of $n ((L \cap M) \cup N)$ .

31. $P$ , $Q$ and $R$ are sets such that $ℰ = P \cup Q \cup R$ . The number of elements in each subset is shown in the Venn diagram.

(a) Find $x$ , given that $n (P) = n (Q)$ .

(b) Find $y$ , given that $n ((P \cup Q)') = n (P \cap Q)$ .

(c) Find $n (ℰ)$ .

32. $ℰ = {x : x is an integer, 2 \leq x \leq 15}$ , $A = {x : 4 < x \leq 13}$ , $P \subset A$ and $P = {x : x is a prime number}$ . List the elements of the following sets:

(a) $A'$ ,

(b) $P$ .

33. $ℰ = {p, q, r, s, t, u, v}$ , $A = {p, q, r, s}$ , $B = {r, t, u, v}$ , $C = {r, s, u, v}$ .

(a) Find $n (A \cap C)$ .

(b) List the elements of (i) $(B \cup C)'$ , (ii) $(A \cup C) \cap B$ .

34. $ℰ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}$ , $A = {x : x is a factor of 12}$ , $B = {x : 2 x - 5 < 3}$ . List the elements of the set

(a) $A$ ,

(b) $B$ ,

(c) $A \cap B'$ .

35. (a) It is given that $n (ℰ) = 50$ and that $P$ and $Q$ are two sets for which $n (P \cap Q) = 6$ , $n (P) = 18$ , $n (Q) = 14$ .

(i) Draw a Venn diagram to illustrate this information.

(ii) Find $n (P' \cap Q')$ .

(b) Copy this Venn diagram and add to your diagram a set $N$ which is such that $N \subset L$ and $N \cap M = \emptyset$ .

36. $ℰ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}$ , $A = {x : x is a factor of 18}$ , $B = {x : 3 x - 1 > 20}$ . List the elements of the sets (a) $A$ , (b) $B$ , (c) $A' \cap B$ .

37. (a) On the Venn diagram in the answer space shade $P \cup (Q \cap R)$ .

(b) There are 30 boys in a class. Of these, 22 play football, 17 play cricket and 3 play neither football nor cricket. It is given that $ℰ = {boys in the class}$ , $F = {boys who play football}$ , and $C = {boys who play cricket}$ .

(i) Find $n (F \cap C)$ .

(ii) Express in set notation ${boys who play cricket but not football}$ .

38. It is given that $A = {2, 3, 4, 6, 8, 9, 10, 12}$ , $B = {3, 5, 7, 9, 11}$ , and $ℰ = A \cup B$ .

(a) List the elements of the set $A \cap B$ .

(b) Find $n (ℰ)$ .

(c) If one element of $ℰ$ is chosen at random, what is the probability that it is a member of both $A$ and $B$ ?

39. $ℰ = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}$ , $P = {x : x is a multiple of 2}$ , $Q = {x : x is a multiple of 3}$ , $R = {x : x is a multiple of 5}$ .

(a) List the elements of (i) $P \cap R$ , (ii) $Q \cap (P \cup R)$ .

(b) Find $n (Q' \cap R)$ .

40. Some people were interviewed to find out whether they spoke French, Spanish, French and Spanish, or neither French nor Spanish. In the Venn diagram, $ℰ$ is the set of people who were interviewed, $F$ is the set of people who spoke French, and $S$ is the set of people who spoke Spanish. The letters $w$ , $x$ , $y$ and $z$ represent the number of people in each of the subsets shown.

Given that $n (ℰ) = 250$ , $n (F) = 80$ and $n (S) = 220$ , find

(a) the maximum possible value of $w$ ,

(b) the maximum possible value of $z$ ,

(c) the maximum possible number of people who spoke both French and Spanish.

41. $ℰ = {7, 8, 9, 10, 11, 12, 13, 14, 15}$ , $A = {x : x is a multiple of 3}$ , $B = {x : x is an odd number}$ , $C = {x : 10 \leq x \leq 13}$ .

(a) Find $n (A \cup B)$ .

(b) List the elements of the set $C'$ .

(c) Find the element $x$ such that $x \in (B \cap C')$ and $x \notin A$ .

42. $ℰ = {4, 5, 6, 7, 8, 9, 10, 11, 12}$ , $P = {x : x is a multiple of 3}$ , $Q = {x : x is even}$ , $R = {x : 6 < x \leq 11}$ .

(a) List the elements of set $R'$ .

(b) Find $n (P \cap Q)$ .

(c) Find the element $x$ such that $x \in Q \cap R'$ and $x \notin P$ .

43. (a) On the Venn Diagram in the answer space shade the set $A \cup (B \cap C)$ .

(b) There are 28 girls in a class. Of these 17 sing in the choir and 15 play the piano. It is given that $ℰ = {girls in the class}$ , $S = {girls who sing in the choir}$ , $P = {girls who play the piano}$ .

(i) Find the smallest possible value of $n (S \cap P)$ .

(ii) Express in set notation ${Girls who neither sing in the choir nor play the piano}$ .

44. (a) $ℰ = {3, 4, 5, 6, 7, 8, 9, 10, 11}$ , $P = {x : x is a factor of 12}$ , $Q = {x : x is an odd integer}$ .

List the elements of the set (i) $P$ , (ii) $(P \cup Q)'$ .

(b) Express in set notation, as simply as possible, the subset shaded in the Venn Diagram.

45. $P$ , $Q$ and $R$ are sets such that $ℰ = P ∪ Q \cup R$ . The numbers of elements in some of the subsets are shown in the Venn Diagram. Given that $n (ℰ) = 40$ , find

(a) $n (Q \cap R)$ ,

(b) $n (P \cup Q)$ ,

(c) $n (P' \cap R)$ .

46. (a) $ℰ = {2, 3, 4, 5, 6, 7, 8, 9}$ , $L = {x : x is a factor of 20}$ and $M = {x : x is an even integer}$ .

List the elements of the set (i) $L$ , (ii) $(L \cup M)'$ .

(b) Express in set notation, as simply as possible, the subset shaded in the Venn Diagram.

47. Each of a group of 20 students studies at least one of the three subjects Chemistry, Physics and Biology.

All those who study Physics also study Chemistry.

3 students study all three subjects.

4 students study only Chemistry.

8 students study Physics.

14 students study Chemistry.

(a) Draw a Venn diagram to illustrate this information.

(b) How many students study only Biology?

(c) How many students study Chemistry and Biology but not Physics?

48. People staying at a holiday hotel are able to take part in Sailing, Swimming and Golf.

4 people take part in all three activities.

17 people take part in Sailing and Swimming but not Golf.

21 people take part in Swimming and Golf but not Sailing.

12 people take part in Sailing and Golf but not Swimming.

$x$ people take part in Swimming only.

42 people take part in Sailing only.

$(x - 2)$ people take part in Golf only.

16 people do not take part in any of these activities.

(a) Copy the given Venn diagram and on your copy show the number of people in each subset.

(b) Given that 250 people are staying at the hotel, calculate

(i) $x$ ,

(ii) the number of people who do not take part in Swimming.

49. (a) On the Venn diagram in the answer space, shade the set $A \cup B'$ .

(b) $ℰ = {x : x is a positive integer}$ , $P = {x : x < 9}$ , $Q = {x : x \geq 4}$ .

(i) List the elements of $P \cap Q$ .

(ii) Find $n (Q')$ .

50. By drawing a Venn diagram, or otherwise, answer the following questions.

(a) Given that $n (ℰ) = 60$ , $n (S) = 33$ , and $n (F) = 36$ , find the least possible value of $n (S \cap F)$ .

(b) In a group of 60 people, 33 can speak Spanish and 36 can speak French. Find the greatest possible number of people who can speak Spanish, but not French.

51. By drawing a Venn diagram, or otherwise, answer the following questions.

(a) Given that $n (ℰ) = 50$ , $n (C) = 33$ and $n (M) = 28$ , find the least possible value of $n (C \cap M)$ .

(b) In a group of 50 people, 33 can speak Chinese and 28 can speak Malay. Find the greatest possible number of people who can speak Chinese but not Malay.

52. $ℰ = {x : 1 \leq x \leq 30}$ , $A = {x : x is a multiple of 4}$ , $B = {x : x is a multiple of 3}$ , $C = {x : x is a multiple of 12}$ .

(a) List the elements of the set $A$ .

(b) Find $n (A \cap B′)$ .

(c) Write down in set notation an equation involving the three sets, $A$ , $B$ and $C$ .

53. In the Venn Diagram shown in the answer space, $ℰ$ represents the set of all triangles, $I$ represents the set of Isosceles triangles and $R$ represents the set of Right-angled triangles.

(a) Add the set $E$ to the Venn Diagram where $E$ represents the set of Equilateral triangles.

(b) A triangle has sides 3 m, 4 m and 5 m. On the diagram mark and label a point $T$ , to represent this triangle.

54. In this question $F_{p}$ stands for the set of factors of the number $p$ . For example, $F_{12} = {1, 2, 3, 4, 6, 12}$ , and the number of elements in $F_{12}$ is $n (F_{12}) = 6$ .

(a) List the elements of $F_{18}$ .

(b) Show that $n (F_{30}) = 8$ .

(c) Given that $n (F_{p}) = 2$ , what kind of number is $p$ ?

In the remaining parts of the question there are many possible answers but you only need to give one for each.

(d) Find a value of $p$ such that $F_{p} \subset F_{12}$ .

(e) Find a value of $p$ such that $n (F_{p}) = 3$ .

55. All members of a Sports Club were asked whether they played cricket or tennis. The survey produced the following three pieces of information: 35 members played cricket, 27 members played tennis, three times as many members played both cricket and tennis as played neither.

(a) Take $x$ to be the number of members who played neither sport. Using a Venn diagram, or otherwise, find in terms of $x$ , in their simplest form, expressions for the number of members who played

(i) both sports,

(ii) one or other of the two sports but not both.

(b) Given that there are 52 members in the Club, find how many played neither sport.

56. There are 30 students in a class. 22 study Mathematics, 16 study Science and 2 study neither Mathematics nor Science. By drawing a Venn diagram, or otherwise, find the number of students who study both Mathematics and Science.

57. There are 120 students in the third form of a school. 40 of them study Woodwork, 28 of them study Metalwork and 6 of them study both subjects. By drawing a Venn diagram, or otherwise, find the number of students who study neither Woodwork nor Metalwork.

58. Example 1

The number line above shows the set $P$ of real numbers $x$ , where $P = {x : x \geq 4}$ .

Example 2

The number line above shows the set $Q$ of real numbers $x$ , where $Q = {x : 2 \leq x < 5}$ .

On the number lines given in the answer spaces below, illustrate the following sets of real numbers $x$ :

(a) $A = {x : x \leq 3}$ ,

(b) $B = {x : - 2 < x < 6}$ ,

(c) $A \cap B$ .

59. There are 24 children on a school outing.

At lunchtime 11 of them ate a sandwich,

9 of them ate a banana,

$n$ of them ate neither a sandwich nor a banana.

By drawing Venn diagrams, or otherwise, find

(a) the smallest possible value of $n$ ,

(b) the largest possible value of $n$ .

60. There are 26 boys in a class at school. 14 of them play football, 10 of them play cricket and $n$ of them play neither football nor cricket. By drawing a Venn diagram, or otherwise, find

(a) the smallest possible value of $n$ ,

(b) the largest possible value of $n$ .

61. $A$ , $B$ and $C$ are subsets of the universal set $ℰ$ where $A \subset B$ , $A \cap C = \emptyset$ , $B \cap C \neq \emptyset$ and $C$ is not a subset of $B$ . Show one way in which the sets $A$ and $C$ can be added to the Venn Diagram.

62. (a) On the Venn diagram, shade the set $B' \cap T$ .

(b) Members of a sports club can play either badminton or table tennis. There are 30 members in the club. It is known that 24 members play badminton, 16 play table tennis and 4 do not play either sport. Find how many play table tennis but not badminton.

63. (a) On the Venn diagram in the answer space, shade the set $(A \cap B) \cup C$ .

(b) The Universal set is the set of all positive integers and $P = {x : x < 10}$ . Complete the statement: $P' = {x :}$ .

64. (a) $A$ , $B$ and $C$ are subsets of the universal set $ℰ$ , $B \cup A = B$ and $B \cap C = \emptyset$ . Illustrate this on the Venn diagram in the answer space.

(b) In a group of people, 15 drink tea, 28 drink coffee, 12 drink both tea and coffee and 6 drink neither tea nor coffee. By drawing a Venn diagram, or otherwise, find the total number of people in the group.

65. $ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}$ , $A = {x : x is an even number}$ , $B = {x : x is a multiple of 3}$ .

(a) Draw a Venn Diagram to illustrate these sets, indicating the members of each subset.

(b) (i) A number $n$ is chosen at random from $ℰ$ . Find the probability that

(a) $n \in A$ ,

(b) $n \in A'$ ,

(c) $n \in A \cap B$ ,

(d) $n = 14$ .

(ii) A second number $m$ is chosen at random from the complete set $ℰ$ . Find the probability that the sum of $n$ and $m$ is 18.

66. (a) On the Venn Diagram, shade the set $(P \cup R) \cap Q$ .

(b) Express in set notation, as simply as possible, the set shaded in the Venn Diagram.

(c) There are 27 people in a restaurant. Of these, 10 eat fish, 15 eat meat and 9 eat neither fish nor meat. Using a Venn Diagram, or otherwise, find the number of people in the restaurant who eat fish but not meat.