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

R3.1.Four options are given against each statement. Encircle the correct option. (i) The set builder form of {1, 1/3, 1/5, 1/7, ...} is: (a) {x|x=1/n, n∈W} (b) {x|x=1/(2n+1), n∈W} (c) {x|x=1/(n+1), n∈W} (d) {x|x=2n+1, n∈W}. (ii) If A={ }, then P(A) is: (a){ } (b){1} (c){{ }} (d)φ. (iii) If U={1,2,3,4,5}, A={1,2,3} and B={3,4,5}, then U-(A∩B) is: (a){1,2,4,5} (b){2,3} (c){1,3,4,5} (d){1,2,3}. (iv) If A and B are overlapping sets, then n(A-B) is equal to (a) n(A) (b) n(B) (c) A∩B (d) n(A)-n(A∩B). (v) If A⊆B and B-A≠φ, then n(B-A) is equal to (a) 0 (b) n(B) (c) n(A) (d) n(B)-n(A). (vi) If n(A∪B)=50, n(A)=30 and n(B)=35, then n(A∩B)=: (a)23 (b)15 (c)9 (d)40. (vii) If A={1,2,3,4} and B={x,y,z}, then cartesian product of A and B contains exactly ___ elements. (a)13 (b)12 (c)10 (d)6. (viii) If f(x)=x^2-3x+2, then the value of f(a+1) is equal to: (a) a+1 (b) a^2+1 (c) a^2+2a+1 (d) a^2-a. (ix) Given that f(x)=3x+1, if f(x)=28, then the value of x is: (a)9 (b)27 (c)3 (d)18. (x) Let A={1,2,3} and B={a,b} two non-empty sets and f:A→B be a function defined as f={(1,a),(2,b),(3,b)}, then which of the following statement is true? (a) f is injective (b) f is surjective (c) f is bijective (d) f is into only

Result

(i) (b) {xmid x=12n+1, nin W}

(ii) (c) {{ }}

(iii) (a) {1,2,4,5}

(iv) (d) n(A)-n(Acap B)

(v) (d) n(B)-n(A)

(vi) (b) 15

(vii) (b) 12

(viii) (d) a²-a

(ix) (a) 9

(x) (b) f is surjective

R3.2.Write each of the following sets in tabular forms: (i) {x|x=2n, n∈N} (ii) {x|x=2m+1, m∈N} (iii) {x|x=11n, n∈W∧n<11} (iv) {x|x∈E∧4<x<6} (v) {x|x∈O∧5<x<7} (vi) {x|x∈Q∧x^2=2} (vii) {x|x∈Q∧x=-x} (viii) {x|x∈R∧x∉Q'}

Result

(i) {2,4,6,8,10,dots}

(ii) {3,5,7,9,11,dots}

(iii) {0,11,22,33,44,55,66,77,88,99,110}

(iv) varphi

(v) varphi

(vi) varphi

(vii) {0}

(viii) Q

R3.3.Let U={1,2,3,4,5,6,7,8,9,10}, A={2,4,6,8,10}, B={1,2,3,4,5} and C={1,3,5,7,9}. List the members of each of the following sets: (i) A' (ii) B' (iii) A∪B (iv) A-B (v) A∩C (vi) A'∪C' (vii) A'∪C (viii) U'

R3.4.Using the Venn diagrams, if necessary, find the single sets equal to the following: (i) A' (ii) A∩U (iii) A∪U (iv) A∪φ (v) φ∩φ

Result

(i) A'=U-A

(ii) Acap U=A

(iii) Acup U=U

(iv) Acupvarphi=A

(v) varphicapvarphi=varphi

R3.5.Use Venn diagrams to verify the following: (i) A-B=A∪B' (ii) (A-B)'∩B=B

(i) Venn diagrams for overlapping and disjoint A,B show the shaded regions for A-B and Acup B' differ

Result

(i) A-Bne Acup B'

(ii) Venn diagrams for Acap B=varphi and Acap Bnevarphi both show the shaded region for (A-B)'cap B equals B

(ii) (A-B)'cap B=B

R3.6.Verify the properties for the sets A, B and C given below: (i) Associativity of Union (ii) Associativity of intersection (iii) Distributivity of Union over intersection (iv) Distributivity of intersection over union, for (a) A={1,2,3,4}, B={3,4,5,6,7,8}, C={5,6,7,9,10} (b) A=φ, B={0}, C={0,1,2} (c) A=N, B=Z, C=Q

(a)(i) Acup(Bcup C)={1,2,3,4}cup{3,4,5,6,7,8,9,10}={1,dots,10}=(Acup B)cup C

(a)(ii) Acap(Bcap C)={1,2,3,4}cap{5,6,7}=varphi=(Acap B)cap C

(a)(iii) Acup(Bcap C)={1,2,3,4}cup{5,6,7}={1,2,3,4,5,6,7}=(Acup B)cap(Acup C)

Result

(a)(iv) Acap(Bcup C)={1,2,3,4}cap{3,4,5,6,7,8,9,10}={3,4}=(Acap B)cup(Acap C)

(b) A=varphi,B={0},C={0,1,2}: all four properties verified, e.g. Acup(Bcap C)={0}=(Acup B)cap(Acup C)

(c) A=N,B=Z,C=Q: all four properties verified using Nsubseteq Zsubseteq Q, e.g. Acup(Bcap C)=Ncup Z=Z=(Acup B)cap(Acup C)

R3.7.Verify De Morgan's Laws for the following sets: U={1,2,3,...,20}, A={2,4,6,...,20} and B={1,3,5,...,19}.

A'={1,3,5,dots,19}; B'={2,4,6,dots,20}

Result

(A∪B)'=A'∩B' Acup B=U ⇒ (Acup B)'=varphi=A'cap B'

(A∩B)'=A'∪B' Acap B=varphi ⇒ (Acap B)'=U=A'cup B'

R3.8.Consider the set P = {x| x=5m, m∈N} and Q={x| x=2m, m∈N}. Find P∩Q.

Given

P={5,10,15,20,25,dots}, Q={2,4,6,8,10,12,dots}

Result

Pcap Q={10,20,30,40,50,dots}={xmid x=10m, min N}

R3.9.From suitable properties of union and intersection, deduce the following results: (i) A∩(A∪B)=A∪(A∩B) (ii) A∪(A∩B)=A∩(A∪B)

(i) Acap(Acup B)=(Acap A)cup(Acap B)=Acup(Acap B)

(ii) Acup(Acap B)=(Acup A)cap(Acup B)=Acap(Acup B)

Acap(Acup B)=Acup(Acap B); Acup(Acap B)=Acap(Acup B)

R3.10.If g(x) = 7x - 2 and s(x) = 8x^2 - 3 find: (i) g(0) (ii) g(-1) (iii) g(-5/3) (iv) s(1) (v) s(-9) (vi) s(7/2)

Working

(i) g(0)=7(0)-2

Result

(i) g(0)=-2

Working

(ii) g(-1)=7(-1)-2=-7-2

Result

(ii) g(-1)=-9

Working

(iii) gleft(-53right)=7left(-53right)-2=-353-2

Result

(iii) gleft(-53right)=-413

Working

(iv) s(1)=8(1)²-3=8-3

Result

(iv) s(1)=5

Working

(v) s(-9)=8(-9)²-3=648-3

Result

(v) s(-9)=645

Working

(vi) sleft(72right)=8left(72right)²-3=98-3

Result

(vi) sleft(72right)=95

R3.11.Given that f(x) = ax + b, where a and b are constant numbers. If f(-2) = 3 and f(4) = 10, then find the values of a and b.

Working

f(-2)=-2a+b=3 quad (i)

f(4)=4a+b=10 quad (ii)

(ii)-(i): 6a=7 ⇒ a=76

Result

b=10-4left(76right)=163

R3.12.Consider the function defined by k(x) = 7x - 5. If k(x) = 100, find the value of x.

Working

7x-5=100

7x=105

Result

x=1057

R3.13.Consider the function g(x) = mx^2+n, where m and n are constant numbers. If g(4) = 20 and g(0) = 5, find the values of m and n.

Working

g(0)=m(0)²+n=5 ⇒ n=5

g(4)=16m+n=20

16m+5=20 ⇒ 16m=15

Result

m=1516

R3.14.A shopping mall has 100 products from various categories labeled 1 to 100, representing the universal set U. The products are categorized as follows: Set A: Electronics, consisting of 30 products labeled from 1 to 30. Set B: Clothing comprises 25 products labeled from 31 to 55. Set C: Beauty Products, comprising 25 products labeled from 76 to 100. Write each set in tabular form, and find the union of all three sets.

Given

U={1,dots,100}; A={1,dots,30}; B={31,dots,55}; C={76,dots,100}

Result

Acup Bcup C={1,dots,30,31,dots,55,76,dots,100}

R3.15.Out of the 180 students who appeared in the annual examination, 120 passed the math test, 90 passed the science test, and 60 passed both the math and science tests. (a) How many passed either the math or science test? (b) How many did not pass either of the two tests? (c) How many passed the science test but not the math test? (d) How many failed the science test?

Given

Total=180, Math=120, Science=90, Both=60

Formula

(a) MathcupScience=Math+Science-Both

Result

(a) 120+90-60=150

(b) 180-150=30

(c) 90-60=30

(d) 180-90=90

R3.16.In a software house of a city with 300 software developers, a survey was conducted to determine which programming languages are liked more. The survey revealed the following statistics: 150 developers like Python, 130 developers like Java, 120 developers like PHP, 70 developers like both Python and Java, 60 developers like both Python and PHP, 50 developers like both Java and PHP, 40 developers like all three languages: Python, Java and PHP. (a) How many developers use at least one of these languages? (b) How many developers use only one of these languages? (c) How many developers do not use any of these languages? (d) How many developers use only PHP?

Given

n(P)=150, n(J)=130, n(H)=120, n(Pcap J)=70, n(Pcap H)=60, n(Jcap H)=50, n(Pcap Jcap H)=40

Formula

(a) n(Pcup Jcup H)=n(P)+n(J)+n(H)-n(Pcap J)-n(Pcap H)-n(Jcap H)+n(Pcap Jcap H)

Result

(a) n(Pcup Jcup H)=150+130+120-70-60-50+40=260

(b) only Python 150-70-60+40=60

(b) only Java 130-70-50+40=50

(b) only PHP 120-60-50+40=50

(b) 60+50+50=160

(c) 300-260=40

(d) n(H)=50

This should read 'only-PHP = 50', not n(H)=50 — n(H) itself is 120 (given). The numeric answer 50 is still correct because it equals the only-PHP figure already computed in part (b).

Unit 3: Sets and Functions — Review Exercise