SETS

WORKSHEETS

Name _______________________________________ Date _______________ Period ______

IBMS VENN DIAGRAMS

Sets 1. A survey of 80 sophomores at a certain western college showed the following: 36 take English 32 take History 32 take Political Science 16 take History and English 16 take Political Science and History 14 take Political Science and English 6 take all three How many students:

(a) take English and neither of the other two? (b) take none of the three courses? (c) take History, but neither of the other two? (d) take Political Science and History but not English? (e) do not take Political Science?

2. The following list shows the preferences of 102 people at a wine-tasting party: 99 like Spañada 94 like Ripple and Boone’s 96 like Ripple 96 like Spañada and Boone’s 99 like Boone’s Farm Apple Wine 93 like all three 95 like Spañada and Ripple How many people prefer:

(a) none of the three? (b) Spañada, but not Ripple? (c) Anything but Boone’s Farm? (d) only Ripple? (e) Exactly two kinds of wine?

3. Toward the middle of the harvesting season, peaches for canning come in three types: earlies, lates,

and extra lates, depending on the expected date of ripening. During a certain week, the following data were recorded at a small fruit-receiving station:

34 trucks went out carrying early peaches 30 had lates and extra lates 61 had late peaches 8 had earlies and extra lates 50 had extra lates 6 had all three 25 had earlies and lates 9 had only figs (no peaches at all)

(a) How many trucks had only late variety peaches? (b) How many had only extra lates? (c) How many had only one variety of peaches? (d) How many trucks in all went out during the week?

Name _________________________________________Date _______________ Period ______

IBMS SET NOTATION

Sets Find ( )n A for the following sets. 1. { }1,0,1, 2,3, 4,5A = − 2. { }1, 2,3, ,1000… 3. { }: is a state in the U.S.A x x= List the elements of each set. 4. { }: is a natural number less than 6x x 5. {all prime numbers less than 30} Write the following in set builder notation. There may be more than one way to do this. 6. { }7,9,11,13,15,17 7. { }10, 20,30, 40,… 8. { }25,50,75,100,125,… 9. {January, February, … , December} Let U { }, , , , , , , , ,1 2 3 4 5 6 7 8 9 10= . Find the complement of the following sets. 10. { }1, 4,6,8 11. { }2,5,7,9,10 12. { }3,5,7,9 13. {Prime numbers} 14. U 15. ∅ Tell whether each statement is true or false. Let U = {2, 4, 6, 8, 10, 12, 14} A = {2, 4, 6, 8, 10, 12} B = {2, 4, 8, 10} C = {4, 10, 12} D = {2, 10} 16. 4 C∈ 17. 7 A∈ 18. 6 B∉ 19. ∅∉ U

20. A⊂ U 21. D B⊆ 22. A B⊂ 23. B∅⊂ 24. ∅⊆∅ 25. D B⊄ 26. A C⊄ 27. A′ ⊂ U

Name __________________________________________ Date _______________ Period ______

IBMS INTERSECTION/UNION OF SETS

Sets Perform the indicated operations. Let U = {2, 3, 4, 5, 7, 9} X = {2, 3, 4, 5) Y = {3, 5, 7, 9} Z = {2, 4, 5, 7, 9} 1. X Y∩ 2. X Y∪ 3. Y Z∪ 4. Y Z∩ 5. X U∪ 6. Y U∩ 7. X ′ 8. Y ′ 9. X Y′ ′∩ 10. X Z′∩ 11. Z ′∩∅ 12. Y ′∪∅ 13. ( )X Y Z∪ ∩ 14. ( )Y X Z∩ ∪ 15. ( )Y Z X′∪ ∪

16. ( )X Y Z′ ′∪ ∪ 17. ( )Z X Y′′∪ ∩ 18. ( )Y X Z′′ ′∩ ∪ Use the sets above to determine if each equation is true or false.

19. X Y Y X∪ = ∪ 20. ( )X X′′ = 21. X ∪∅ =∅ 22. Y Y∪∅ = 23. X ∩∅ =∅ 24. ( ) ( )X Y Z X Y Z∪ ∪ = ∪ ∪

25. ( ) ( )X Y Z X Y Z∩ ∩ = ∩ ∩ 26. ( )X Y X Y ′′ ′∩ = ∩

27. ( )X Y X Y ′′ ′∪ = ∪ 28. ( )X Y X Y ′′ ′∩ = ∪

Name ________________________________________Date _______________ Period ______

IBMS SHADING VENN DIAGRAMS

Sets Use a Venn diagram to shade each of the following sets. 1. ( )A B C∩ ∩ 2. ( )A C B′∪ ∪ 3. ( )A B C′∩ ∪ 4. ( )A B C′∩ ∩ 5. ( )A B C′ ′∩ ∩ 6. ( )A B C∪ ∪ 7. ( )A B C′∩ ∪ 8. ( )A C B′∩ ∩ 9. ( )A B C′∩ ∩

10. ( )A B C′ ′∩ ∪ 11. ( )A B C′ ′ ′∩ ∪ 12. ( )A B C′∩ ∪

A B

C

U

A B

C

U

A B

U

A B

U

BA

C

U

BA

C

U

A B

U

A B

U

Write a description of each shaded area. 13. 14. 15. 16. 17. 18. 19. 20.

Name _________________________________________Date _______________ Period ______

REVIEW FOR SETS TEST 1. List the elements in the set { }is a natural number between 5 and 9x x . 2. Write the set {1, 4, 9, 16, 25, 36, 49} in set-builder notation. 3. Find ( )n A for the set { }400, 401, 402, , 4000A = … . 4. Let { }1, 2, 4,5,a, b, c, d, eU = , find the complement of W if { }1,5,e, d, aW = .

5. Use a Venn diagram to shade the regions representing the set ( )A B C ′′∩ ∩ . 6. In the universal set { }1 10, U x x x= ≤ ≤ ∈ ,

{ }multiples of 2A = ;

{ }multiples of 3B = ;

{ }primesP = .

a) Draw a Venn diagram showing U, A, B and P, clearly writing the numbers in each appropriate region.

b) Perform the indicated operations.

i) ( )A B P′∩ ∩ ii) ( )A P B∪ ∩ 7. Decide whether the following statement is always true or not always true. ( )A B B∩ ⊆ 8. Write a description of each shaded region using , , , , , and A B C ′∩ ∪ as necessary.

A B

C

U

A B

C

U

SETS

NOTEBOOK

Sets Maximum marks will be given for correct answers. Where an answer is wrong, some marks may be given for a correct method provided this is shown by written working. Working may be continued below the box, if necessary. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Incorrect answers with no working will normally receive no marks. M98/530/S(1) 1. The Venn diagram shows the number of players at a sports club who take part in various

sporting activities, where A = {members who do archery}; B = {members who play badminton}; C = {members who take part in cross country}.

Find the number of members who (a) take part in cross country;

(b) take part in more than one activity;

(c) play badminton but do not take part in cross country;

(d) do not do archery.

N02/530/S(1)+ 2. A poll was taken of the leisure time activities of 90 students. 60 students watch TV (T), 60 students read (R), 70 students go to the cinema (C). 26 students watch TV, read and go to the cinema. 20 students watch TV and go to the cinema only. 18 students read and go to the cinema only. 10 students read and watch TV only.

(a) Draw a Venn diagram to illustrate the above information.

(b) Calculate how many students

(i) only watch TV;

(ii) only go to the cinema.

E

C A

B

2

1

3

4

1

8

6

N07/5/MATSD/SP1/ENG/TZ0/XX+ 3. A school offers three activities, basketball (B), choir (C) and drama (D). Every student must

participate in at least one activity. 16 students play basketball only. 18 students play basketball and sing in the choir but do not do drama 34 students play basketball and do drama but do not sing in the choir. 27 students are in choir and do drama but do not play basketball. (a) Enter the above information on the Venn diagram below. 99 of the students play basketball, 88 sing in the choir and 110 do drama. (b) Calculate the number of students x participating in all three activities. (c) Calculate the total number of students in the school. M04/530/S(1)

4. Let 3324, ,1, ,13,26.7,69,103

U π = − −

.

A is the set of all the integers in U. B is the set of all the rational numbers in U.

(a) List all the prime numbers contained in U.

(b) List all the members of A.

(c) List all the members of B.

(d) List all the members of the set A B∩ .

x

B C

D

U

SPEC/530/S(1) 5. The universal set U is defined as the set of positive integers less than 10. The subsets A and B

are defined as: A = { integers that are multiples of 3 }

B = { integers that are factors of 30 }

(a) List the elements of

(i) A ;

(ii) B .

(b) Place the elements of A and B in the appropriate region in the Venn diagram below. M06/5/MATSD/SP1/ENG/TZ0/XX 6. The Venn diagram below shows the universal set of real numbers and some of its important

subsets:

: the rational numbers, : the integers, : the natural numbers. Write the following numbers in the correct position in the diagram.

71,1, , , 3.3333, 316

π−

U

B A

M03/530/S(1) 7. In the Venn diagram below, A, B, and C are subsets of a universal set { }1, 2,3, 4,6,7,8,9U = .

List the elements in each of the following sets. (a) A B∪ (b) A B C∩ ∩ (c) ( )A C B′∩ ∪ N00/530/S(2)

8. [Maximum mark: 14]

In a club with 60 members, everyone attends either on Tuesday for Drama (D) or on Thursday for Sports (S) or on both days for Drama and Sports. One week it is found that 48 members attend for Drama and 44 members attend for Sports and x members attend for both Drama and Sports. (a) (i) Draw and fully label a Venn diagram to illustrate this information. [3 marks]

(ii) Find the number of members who attend for both Drama and Sports. [2 marks]

(iii) Describe, in words, the set represented by ( )D S ′∩ [2 marks]

(iv) What is the probability that a member selected at random attends for Drama only or Sports only? [3 marks]

The club has 28 female members, 8 of whom attend for both Drama and Sports.

(b) What is the probability that a member of the club selected at random

(i) is female and attends for Drama only or Sports only? [2 marks]

(ii) is male and attends for both Drama and Sports? [2 marks]

U B

A C

8

4

3 9

1

7

6

M09/5/MATSD/SP2/ENG/TZ1/XX 9. [Maximum marks : 10]

A survey was carried out in a year 12 class. The pupils were asked which pop groups they like out of the Rockers (R), the Salseros (S), and the Bluers (B). The results are shown in the following diagram.

(a) Write down ( )n R S B∩ ∩ . [1 mark] (b) Find ( )n R′ [2 marks] (c) Describe which group the pupils in the set S B∩ like. [2 marks]

(d) Use set notation to describe the group of pupils who like the Rockers and the Bluers but do not like the Salseros. [2 marks]

There are 33 pupils in the class. (e) (i) Find x. (ii) Find the number of pupils who like the Rockers. [3 marks]

2 0

4

2x

x 53

7

R S

B

U

1 6

8

2

1 43

G T

S

E

N08/5/MATSD/SP1/ENG/TZ0/XX 10. The Venn diagram shows the numbers of pupils in a school according to whether they

study the sciences Physics (P), Chemistry (C) or Biology (B). (a) Write down the number of pupils that study Chemistry only. (b) Write down the number of pupils that study exactly two sciences. (c) Write down the number of pupils that do not study Physics.

(d) Find ( )n P B C∪ ∩ . M99/530/S(1) 11. The sports offered at a retirement village are Golf (G), Tennis (T) and Swimming (S).

The Venn diagram shows the numbers of people involved in each activity.

(a) How many people (i) only play golf? (ii) play both tennis and golf? (iii) do not play golf? (b) Shade the part of the Venn diagram the represents the set G S∩ .

7 8

9

4

5 32

6

P B

C

U

N05/530/S(1) 12. Write down an expression to describe the shaded area on the following Venn diagrams:

(a) (b)

(c) (d) N03/530/S(1) 13. The following Venn Diagram shows the sets U, A, B and C. State whether the following statements are true or false for the information illustrated in the

Venn Diagram. (a) A C∩ =∅ (b) C B C∪ = (c) ( )C A B⊂ ∪ (d) A C′⊂

U A B

C

UA B

UA B

UA B

UA B

C

UA

C

BU

A

C

B

UA

C

B

UA

C

B

N04/530/S(1) 14. Shade the given region on the corresponding Venn Diagram. (a) A B∩ (b) C B∪

(c) ( )A B C ′∪ ∪ (d) A C′∩ SPEC/530/S(1) 15. Given the set of integers, the set of rational numbers, the set of real numbers.

(a) Write down an element that belongs to ∩ .

(b) Write down an element that belongs to ′∩ .

(c) Write down an element that belongs to ′ .

(d) Use a Venn diagram to represent the sets , , and . N01/530/S(1) 16. A committee U has three sub-committees: research R , finance F and purchasing P.

No member belongs to both finance and purchasing sub-committees. Some members belong to both research and purchasing committees. All members of the finance sub-committee also belong to the research sub-committee.

Draw a Venn diagram, showing the relationship between the sets U , R , F and P.

N05/5/MATSD/SP2/ENG/TZ0/XX 17. [Maximum mark: 8] (i) Children in a class of 30 students are asked whether they can swim (S) or ride a bicycle (B). There are 12 girls in the class. 8 girls can swim, 6 girls can ride a bicycle and 4 girls can do both. 16 boys can swim, 13 boys can ride a bicycle and 12 boys can do both. This information is represented in a Venn diagram. (a) Find the values of a and b . [2 marks] (b) Calculate the number of students who can do neither. [2 marks] (c) Write down the probability that a student chosen at random can swim. [2 marks] (d) Given that the student can ride a bicycle, write down the probability that the student is a girl. [2 marks] M05/5/MATSD/SP2/ENG/TZ0/XX 18. The following results were obtained from a survey concerning the reading habits of students. 60% read magazine P 50% read magazine Q 50% read magazine R 30% read magazines P and Q 20% read magazines Q and R 30% read magazines P and R 10% read all three magazines

(a) Represent all of this information on a Venn diagram.

(b) What percentage of students read exactly two magazines?

(c) What percentage of students read at least two magazines?

(d) What percentage of students do not read any of the magazines?

Boys GirlsU

4 a

12 4

b 2

S

B

M07/5/MATSD/SP2/ENG/TZ0/XX 19. There are 49 mice in a pet shop. 30 mice are white (W) 27 mice are male (M) 18 mice have short tails (S) 8 mice are white and have short tails 11 mice are male and have short tails 7 mice have are male but neither white nor short-tailed. 5 mice have all three characteristics 2 have none of the characteristics (a) Using the information above, complete the following Venn diagram. (b) Find (i) ( )n M W∩ (ii) ( )n M S′∪

3

5

S W

M

U

M01/530/S(2) 20. [Maximum mark: 17] The sets A , B and C are subsets of U. They are defined as follows: U = {positive integers less than 16} A = {prime numbers} B = {factors of 36} C = {multiples of 4} (a) List the elements (if any) of the following:

(i) A ;

(ii) B ;

(iii) C ;

(iv) A B C∩ ∩ . [4 marks]

(b) (i) Draw a Venn diagram showing the relationship between the sets U , A , B , and C.

(ii) Write the elements of sets U , A , B , and C in the appropriate places

on the Venn diagram. [4 marks]

(c) From the Venn diagram, list the elements of each of the following (i) ( )A B C∩ ∪

(ii) ( )A B ′∩

(iii) ( )A B C′∩ ∩ [3 marks] (d) Find the probability that a number chosen at random from the universal set U will be

(i) a prime number;

(ii) a prime number, but not a factor of 36;

(iii) a factor of 36 or a multiple of 4, but not a prime number;

(iv) a prime number, given that it is a factor of 36. [6 marks]

M06/5/MATSD/SP1/ENG/TZ0/XX 21. At a certain school there are 90 students studying for their IB diploma. They are required to

study at least one of the subjects: Physics, Biology or Chemistry. 50 students are studying Physics, 60 students are studying Biology, 55 students are studying Chemistry, 30 students are studying both Physics and Biology, 10 students are studying both Biology and Chemistry but not Physics, 20 students are studying all three subjects. Let x represent the number of students who study both Physics and Chemistry but not Biology.

Then 25 x− is the number who study Chemistry only. The figure below shows some of this information and can be used for working.

(a) Express the number of students who study Physics only, in terms of x.

(b) Find x.

(c) Determine the number of students studying at least two of the subjects.

20

1020

Physics Chemistry

Biology

x25 – x

( )with 90U n U =