Sets and Its Applications

1. In a class of 40 pupils , 27 of them play basketball , 25 play tennis while 17 play both . Draw a Venn diagram to represent this information and hence deduce the number of students who play neither game . Ans : 5

2. All the 35 families in a housing estate possess either a car or a motor-cycle , or both . 18 of them have cars and 27 have motor-cycled . Represent these facts in a Venn diagram . How many families have both cars and motor-cycles ? How many of them have cars but not motor-cycles ? Ans : 10 , 8

3. In a recent survey people were asked if the took a vacation in the summer , winter , or spring in the past year . The results were 73 took a vacation in the summer , 51 took a vacation in the winter , 27 took vacation in the spring , and 2 had taken no vacation . Also , 10 had taken vacation at all three times , 33 had taken both a summer and a winter vacation , 18 had taken only

a winter vacation , and 5 had taken both summer and spring but not a winter vacation . (a) How many people had been surveyed ? (b) How many people had taken vacations at exactly two times of the year ? (c) How many people had taken vacations during at most one time of the year ? (d) What percentage had taken vacations during both summer and winter but not spring ? Ans : (a) 107 ; (b) 28 ; (c) 67 ; (d) 22.12%

4. Of the 24 students in a class , 18 like to play basketball and 12 like to play volleyball . Assume x students like to play basketball and volleyball , find the smallest and largest possible value of x .

Ans : 6 , 12

5. In a survey , a number of people were asked about the mode of transport they used on a particular day . Of these questioned , 6 said that they travelled by bus and MRT only . 2 by car and MRT only and 7 by bus ,MRT and car . The number x who travelled by bus only was equal to the number who travelled by bus and car only . Draw a Venn diagram to illustrate the information . (a) Given that 47 people used buses and 36 used MRT , find : (i) value of x , (ii) the number of people who travelled by MRT only , and (iii) the number of people who travelled by at least 2 modes of transport . (b) Given also that a total of 100 people participated in the survey , find the number of people who travelled by car only.

Ans : (a) (i) 17 ; (ii) 21; (iii) 32 ; (b) 30

6. There are 20 students in a chemistry class and 30 students in a physics class . Find the number of

students who are either in physics or in chemistry class ? (i) When the two classes meet at different hours and 10 students are enrolled in both the courses . (ii) Two classes meet at the same hour . Ans : (i) 40 ; (ii) 50

7. Of the 30 people attending a meeting in a conference room , 22 like to use pens and 16 like to use

pencils . ξ represents the set of all the people in the conference room ; B and P represent the sets of people who like to use pens and pencils respectively .

(a) If n ( B∩ P ) = x , draw a Venn diagram to illustrate the given information .

(b) Find , in terms of x , n (B∪P) .

(c) What is the range of the possible values of x ? Ans : (b) 38 – x ; (c) 8 ≤ x ≤ 16

8. In a survey of 100 persons , it was found that 32 read magazine A , 29 read magazine B , 44 read magazine C , 9 read magazines A and B , 11 read magazines A and C , 6 read magazines B and C

and 2 read all the three magazines . Draw a clearly labeled Venn diagram to illustrate all of the given information . Using your Venn diagram , find (a) how many read magazine C only , (b) how many read none of the three magazines . Ans : 29 ; 19

9. In diagram 1 , A , B and C are three sets and the number of elements are as shown. If n(ξ) = 40 ,

and n( ) = 20 , find

(i) the value of x and y ; (ii) n[(B∪C) \ A ] .

10. Each student in a group of 20 studies at least one of the following subjects : Elementary Maths , Additional Maths and Science . All those who study Additional Maths must study Elementary Maths . 2 students study all three subjects . 3 students study only Elementary Maths . 7 students study Additional Maths and 16 students study Elementary Maths . (a) Draw a Venn diagram to illustrate the given information . (b) How many students study Additional Maths but not Science ? (c) How many students study Science only ? (d) How many students study two subjects ? Ans : (b) 5 ; (c) 4 ; (d) 11

A

x 1 2x + 3 B

4C

x-3

y

11. Given that n(ξ) = 23 , n(A∩B) = x , n(A) = y , n(B) = 2y and n( ) = 7 , find the least possible value of y .

12. 某校 1500 位学生中，40%喜爱游泳，50%喜爱看电影，60%喜爱阅读，20%喜爱游泳及 看电影，20%喜爱游泳及阅读，30%喜爱看电影及阅读，10%喜爱这三种爱好。 （i）作一范恩图表示上述资料。 （ii）只有其中一种爱好的学生有多少人？ （iii）只有其中两种爱好的学生有多少人？ （iv）上述任一种爱好都没有的学生有多少人？ （2008 年初中统考） (Ans: 600;600;150)

13. Each of the 35 girls in a class takes part in at least one of the following three activities : Jogging , Swimming and Dancing . Of the 15 girls who choose Jogging , 4 also choose Swimming and Dancing , 2 choose Jogging only , 7 choose Swimming but not Dancing . Of the 20 girls who do not choose Jogging , x choose both Swimming and Dancing , 2x choose only Dancing , 2 choose only Swimming . (a) Draw a Venn diagram to illustrate this information . (b) Find the value of x . (c) How many girls choose Jogging and Dancing but not Swimming ? ( Ans : 6 ; 2 )

14. 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.

15. Some people were interviewed to find out whether they spoke French , Spanish , French and Spanish , or neither French nor Spanish .

In the Venn diagram below , ε F S

ε is the set of people who were interviewed , w x y F is the set of people who spoke French , and z 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(ε)=250 , n(F) = 80 and n(S) = 220 , find (i)the maximum possible value of w , (ii) the maximum possible value of z , (iii) the maximum possible number of people who spoke both French and Spanish. (Ans : 30,30,80)16. 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.

17. There are 50 people on a tour . One day , 26 people went on the morning cruise and 20 to the evening barbecue . Find the least number and greatest number of people who could have gone to both events . (Ans : 5 , 26 )

18. In a class of students , it is found that 15 play violin , 11 play piano , 13 play guitar , 2 students play these three musical instruments,

5 students play violin and piano , 7 students play piano and guitar , 6 students play violin and guitar If they are 7 students do not know play these 3 instruments , find (a) how many students play only violin , piano , guitar ? (b) how many students in the class ? Ans : 1 ; 2 ; 3 ; 30