9th Class Mathematics Unit 3 Sets and Functions Numericals with Solutions

3.2.1.Consider the universal set U = {x: x is multiple of 2 and 0<x≤30}, A = {x: x is a multiple of 6} and B = {x: x is a multiple of 8}. (i) List all elements of sets A and B in tabular form (ii) Find A∩B (iii) Draw a Venn diagram

Given

Sets U={2,4,dots,30}

Result

(i) A={6,12,18,24,30}, B={8,16,24}

(ii) Acap B={24}

(iii) Venn diagram: only-A={6,12,18,30}, Acap B={24}, only-B={8,16}, outside={2,4,10,14,20,22,26,28}

3.2.2.Let U = {x: x is an integer and 0<x≤150}, G = {x: x=2^m for integer m} and H = {x: x is a square}. (i) List all elements of sets G and H in tabular form (ii) Find G∪H (iii) Find G∩H

Result

(i) G={1,2,4,8,16,32,64,128}, H={1,4,9,16,25,36,49,64,81,100,121,144}

(ii) Gcup H={1,2,4,8,9,16,25,32,36,49,64,81,100,121,128,144}

(iii) Gcap H={1,4,16,64}

3.2.3.Consider the sets P = {x: x is a prime number and 0<x≤20} and Q = {x: x is a divisor of 210 and 0<x≤20}. (i) Find P∩Q (ii) Find P∪Q

Given

Sets P={2,3,5,7,11,13,17,19}, Q={1,2,3,5,6,7,10,14,15}

Result

(i) Pcap Q={2,3,5,7}

(ii) Pcup Q={1,2,3,5,6,7,10,11,13,14,15,17,19}

3.2.4.Verify the commutative properties of union and intersection for the following pairs of sets: (i) A={1,2,3,4,5}, B={4,6,8,10} (ii) N, Z (iii) A={x|x∈R∧x≥0}, B=R

(i) Acup B=Bcup A={1,2,3,4,5,6,8,10}; Acap B=Bcap A={4}

(ii) Ncup Z=Zcup N=Z; Ncap Z=Zcap N=N

Result

(iii) Acup B=Bcup A=R; Acap B=Bcap A=A

3.2.5.Let U = {a, b, c, d, e, f, g, h, i, j}, A = {a, b, c, d, g, h}, B = {c, d, e, f, j}. Verify De Morgan's Laws for these sets. Draw Venn diagram.

(A∪B)'=A'∩B' A'={e,f,i,j}, B'={a,b,g,h,i}, Acup B={a,b,c,d,e,f,g,h,j}

Result

(A∪B)'=A'∩B' (Acup B)'={i}=A'cap B'={i}

(A∩B)'=A'∪B' Acap B={c,d}

(A∩B)'=A'∪B' (Acap B)'={a,b,e,f,g,h,i,j}=A'cup B'={a,b,e,f,g,h,i,j}

3.2.6.If U = {1, 2, 3, ..., 20} and A = {1, 3, 5, ..., 19}, verify the following: (i) A∪A'=U (ii) A∩U=A (iii) A∩A'=φ

Given

A'=U-A={2,4,6,dots,20}

Result

(i) Acup A'={1,2,dots,20}=U

(ii) Acap U=A

(iii) Acap A'=varphi

3.2.7.In a class of 55 students, 34 like to play cricket and 30 like to play hockey. Also each student likes to play at least one of the two games. How many students like to play both games?

Given

n(C)=34, n(H)=30, n(U)=55, n(Ccup H)=55

Formula

n(Ccup H)=n(C)+n(H)-n(Ccap H)

Working

55=34+30-n(Ccap H)=64-n(Ccap H)

Result

n(Ccap H)=64-55=9

3.2.8.In a group of 500 employees, 250 can speak Urdu, 150 can speak English, 50 can speak Punjabi, 40 can speak Urdu and English, 30 can speak both English and Punjabi, and 10 can speak Urdu and Punjabi. How many can speak all three languages?

Given

n(Ucup Ecup P)=500, n(U)=250, n(E)=150, n(P)=50, n(Ucap E)=40, n(Ecap P)=30, n(Ucap P)=10

Formula

n(Ucup Ecup P)=n(U)+n(E)+n(P)-n(Ucap E)-n(Ecap P)-n(Ucap P)+n(Ucap Ecap P)

Working

500=250+150+50-40-30-10+n(Ucap Ecap P)=370+n(Ucap Ecap P)

Result

n(Ucap Ecap P)=130

3.2.9.In sports events, 19 people wear blue shirts, 15 wear green shirts, 3 wear blue and green shirts, 4 wear a cap and blue shirts, and 2 wear a cap and green shirts. The total number of people with either a blue or green shirt or cap is 34. How many people are wearing caps?

Given

n(B)=19, n(G)=15, n(Bcap G)=3, n(Bcap C)=4, n(Gcap C)=2, n(Bcup Gcup C)=34, n(Bcap Gcap C)=0

Formula

n(Bcup Gcup C)=n(B)+n(G)+n(C)-n(Bcap G)-n(Bcap C)-n(Gcap C)+n(Bcap Gcap C)

Working

34=19+15+n(C)-3-4-2+0=25+n(C)

Result

n(C)=9

3.2.10.In a training session, 17 participants have laptops, 11 have tablets, 9 have laptops and tablets, 6 have laptops and books, and 4 have both tablets and books. Eight participants have all three items. The total number of participants with laptops, tablets, or books is 35. How many participants have books?

Given

n(L)=17, n(T)=11, n(Lcap T)=9, n(Lcap B)=6, n(Tcap B)=4, n(Lcap Tcap B)=8, n(Lcup Tcup B)=35

Formula

n(Lcup Tcup B)=n(L)+n(T)+n(B)-n(Lcap T)-n(Lcap B)-n(Tcap B)+n(Lcap Tcap B)

Working

35=17+11+n(B)-9-6-4+8=17+n(B)

Result

n(B)=18

3.2.11.A shopping mall has 150 employees labelled 1 to 150, representing the Universal set U. The employees fall into the following categories: Set A: 40 employees with a salary range of 30k-45k, labelled from 50 to 89. Set B: 50 employees with a salary range of 50k-80k, labelled from 101 to 150. Set C: 60 employees with a salary range of 100k-150k, labelled from 1 to 49 and 90 to 100. (a) Find (A'∪B')∩C (b) Find n{A∩(B^c∩C^c)}

Given

U={1,dots,150}; A={50,dots,89}; B={101,dots,150}; C={1,dots,49}cup{90,dots,100}

A'={1,dots,49,90,dots,150}; B'={1,dots,100}; C'={50,dots,89}

(a) A'cup B'={1,dots,150}

Result

(a) (A'cup B')cap C={1,dots,49,90,dots,100}=C

(b) B'cap C'={50,dots,89}; Acap(B'cap C')={50,dots,89}

(b) n{Acap(B^ccap C^c)}=40

3.2.12.In a secondary school with 125 students participate in at least one of the following sports: cricket, football, or hockey. 60 students play cricket. 70 students play football. 40 students play hockey. 25 students play both cricket and football. 15 students play both football and hockey. 10 students play both cricket and hockey. (a) How many students play all three sports? (b) Draw a Venn diagram showing the distribution of sports participation in all the games.

Given

n(Ccup Fcup H)=125, n(C)=60, n(F)=70, n(H)=40, n(Ccap F)=25, n(Fcap H)=15, n(Ccap H)=10

Formula

n(Ccup Fcup H)=n(C)+n(F)+n(H)-n(Ccap F)-n(Fcap H)-n(Ccap H)+n(Ccap Fcap H)

Working

125=60+70+40-25-15-10+n(Ccap Fcap H)

Result

(a) n(Ccap Fcap H)=5

(b) Venn diagram: cricket-only=30, football-only=35, hockey-only=20, Ccap F only=20, Fcap H only=10, Ccap H only=5, all three=5

3.2.13.A survey was conducted in which 130 people were asked about their favourite foods. The survey results showed the following information: 40 people said they liked nihari, 65 people said they liked biryani, 50 people said they liked korma, 20 people said they liked nihari and biryani, 35 people said they liked biryani and korma, 27 people said they liked nihari and korma, 12 people said they liked all three foods nihari, biryani, and korma. (a) At least how many people like nihari, biryani or korma? (b) How many people did not like nihari, biryani, or korma? (c) How many people like only one of the following foods: nihari, biryani, or korma? (d) Draw a Venn diagram.

Given

n(N)=40, n(B)=65, n(K)=50, n(Ncap B)=20, n(Bcap K)=35, n(Ncap K)=27, n(Ncap Bcap K)=12

Formula

(a) n(Ncup Bcup K)=n(N)+n(B)+n(K)-n(Ncap B)-n(Bcap K)-n(Ncap K)+n(Ncap Bcap K)

Result

(a) n(Ncup Bcup K)=40+65+50-20-35-27+12=85

(b) 130-85=45

(c) only nihari 40-20-27+12=5

(c) only biryani 65-20-35+12=22

(c) only korma 50-27-35+12=0

(c) 5+22+0=27

(d) Venn diagram: nihari-only=5, Ncap B only=8, biryani-only=22, Ncap K only=15, all three=12, Bcap K only=23, korma-only=0, outside=45

Unit 3: Sets and Functions — Exercise 3 2