Q: List all the elements of the following sets:(i) A={x:x is an odd natural number}(ii) B={x:x is an integer,2−1<x<29}(iii) C={x:x is an integer,x2≤4}(iv) D={x:x is a letter in the word LOYAL}(v) E={x:x is a month of a year not having 31 days}(vi) F={x:x is a consonant in the English alphabet which precedes k}

(i) A={x:x is an odd natural number} So the elements are A={1,3,5,7,9……} (ii) B={x:x is an integer, −1/2 4 Here02=0≤4,12=1≤4,22=4≤4,32=9>4 So we getC={−2,−1,0,1,2} (iv) D = {x: x is a letter in the word "LOYAL"}So the elements are D={L,O,Y,A} (v) E = {x: x is a month of a year not having 31 days }So the elements are E={February,April,June,September,November} (vi) F={x:x is a consonant in the English alphabet which proceeds k} So the elements are F={b,c,d,f,g,h,j}.

Question 1

Which of the following are sets ? Justify your answer.(i) The collection of all the months of a year beginning with the letter J.(ii) The collection of ten most talented writers of India.(iii) A team of eleven best-cricket batsmen of the world.(iv) The collection of all boys in your class.(v) The collection of all natural numbers less than 100.(vi) A collection of novels written by the writer Munshi Prem Chand.(vii) The collection of all even integers.(viii) The collection of questions in this Chapter.(ix) A collection of most dangerous animals of the world

Accepted Answer

(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identify a month that belongs to this collection.Hence, this collection is a set.(ii) The collection of ten most talented writers of India is not a well-defined collection because the criteria for deterring a writer's talent may vary from person to person.Hence, this collection is not a set.(iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for deterring a batsman's  talent may vary from person to person.Hence, this collection is not a set.(iv) The collection of all boys in your class is a well-defined collection because, you can definitely identify a boy who belongs to this collection.Hence, this collection is a set.(v) The collection of  all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection.Hence, this collection is a set.(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection.Hence, this collection is a set.(vii) The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection.Hence, this collection is a set.(viii) The collection  of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter.Hence, this collection is a set.(ix) The collection of most dangerous animals of the world is not a well-defined collection because that criteria for detering the dangerousness of an animal can vary from animal to animal.Hence, this collection is not a set.

Question 2

Let A={1,2,3,4,5,6}. Insert the appropriate symbol ∈ or ∈ in the blank  spaces:(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A (iv) 4. . . A (v) 2. . .A (vi) 10. . .A

Accepted Answer

(i) 5 ∈A (ii) 8∈/A (iii) 0∈/A (iv) 4∈A (v) 2∈A (vi) 10∈/A

Question 3

Write the following sets in roster form:(i) A={x:x is an integer and−3<x<7}(ii) B={x:x is a natural number less than 6}(iii) C={x:x is a two-digit natural number such that the sum of its digits is 8}(iv) D={x:x is a prime number which is divisor of 60}(v) E= The set of all letters in the word TRIGONOMETRY(vi) F= The set of all letters in the word BETTER

Accepted Answer

(i) A={x:x is an integer and−3<x<7} As per the condition, -2,-1,0,1,2,3,4,5, and 6 only are the elements of this set. Hence, the given set can be written in roster form as A={−2,−1,0,1,2,3,4,5,6}  >(ii) B={x:x  is a natural number less than 6}The elements of this set are 1,2,3,4 and 5 only.Therefore, the given set can be written in roster form as B={1,2,3,4,5}(iii) C={x.x  is a two-digit natural number such that the sum of its digits is 8}The elements of this set are 17,26,35,44,53,62,71 and 80 only,Therefore, this set can be written in roster form asC={17,26,35,44,53,62,71,80}(iv) D = {x: x is a prime number which is a divisor of 60}∴60=2×2×3×5 The elements of this set are 2,3 , and 5 only.Therefore, this set can be written in roster form as D={2,3,5}.

Question 4

Write the following sets in the set-builder form(i) {3,6,9,12} (ii) {2,4,8,16,32} (iii) {5,25,125,625} (iv) {2,4,6,...} (v) {1,4,9,...,100}

Accepted Answer

(i) {3,6,9,12}={x:x=3n,nϵN and 1≤n≤4}Alternatively, {x:x is a multiple of 3 and x≤12}(ii) {2,4,8,16,32}It can be seen that 2=21,4=22,8=23,16=24, and 32=25.∴{2,4,8,16,32}={x:x=2n,nϵN and 1≤n≤5}(iii) {5,25,125,625}It can be seen that 5=51,25=52,125=53, and 625=54.∴{5,25,125,625}={x:x=5n,nϵN and  1≤n≤4}(iv) {2,4,6,....}It is a set of all even natural numbers.∴{2,4,6,....}={x:x is an even natural number}(v) {1,4,9,.....100}∴{1,4,9...100}={x:x=n2,nϵN and 1≤n≤10}Alternatively, {x:x is a square of natural number and x≤100}

Question 5

List all the elements of the following sets:(i) A={x:x is an odd natural number}(ii) B={x:x is an integer,2−1​<x<29​}(iii) C={x:x is an integer,x2≤4}(iv) D={x:x is a letter in the word LOYAL}(v) E={x:x is a month of a year not having 31 days}(vi) F={x:x is a consonant in the English alphabet which precedes k}

Accepted Answer

(i) A={x:x is an odd natural number} So the elements are A={1,3,5,7,9……} (ii) B={x:x is an integer, −1/24 Here02=0≤4,12=1≤4,22=4≤4,32=9>4 So we getC={−2,−1,0,1,2} (iv) D = {x: x is a letter in the word "LOYAL"}So the elements are D={L,O,Y,A} (v) E = {x: x is a month of a year not having 31 days }So the elements are E={February,April,June,September,November} (vi) F={x:x is a consonant in the English alphabet which proceeds k} So the elements are F={b,c,d,f,g,h,j}.

Question 6

Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

Accepted Answer

(i) All the elements of this set are natural number as well as the divisors of 6.Therefore, (i) matches with (c).(ii) It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.Therefore, (ii) matches with (a).(iii) All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).(iv) All the elements of this set are odd natural numbers less than 10. Therefore, (iv) matches with (b).

Question 7

Which of the following are examples of the null set?(i) Set of odd natural numbers divisible by 2(ii) Set of even prime numbers.(iii) {x:x is a natural numbers, x 7} (iv) {y:y is a point common to any two parallel lines}

Accepted Answer

(i) A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.(ii) A={2}.A set of even prime numbers is not a null set because 2 is an even prime number.(iii) {x:x is a natural number,x<5  and x>7} is a null set because a number cannot be simultaneously less than 5 and greater than 7.(iv) {y:y is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.

Question 8

Which of the following sets are finite or infinite(i) The set of months of a year(ii) {1,2,3,....}(iii) {1,2,3,....99,100}(iv) The set of positive integers greater than 100(v) The set of prime numbers less than 99

Accepted Answer

(i) The set of months of a year is a finite set because it has 12 elements.(ii) {1,2,3....} is an infinite set as it has infinite number of natural numbers.(iii) {1,2,3,...99,100} is a finite set because the numbers from 1 to 100 are finite in number.(iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.(v) The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.

Question 9

In the following, state whether A=B or not:(i) A={a,b,c,d}B={d,c,b,a}(ii) A={4,8,12,16}B={8,4,16,18}(iii) A={2,4,6,8,10}B={x:x is a positive even integer and x≤10}(iv) A={x:x is a multiple of 10},B={10,15,20,25,30,...}

Accepted Answer

(i) A={a,b,c,d};B={d,c,b,a}The order in which the elements of a set are listed is not significant.∴A=B(ii) A={4,8,12,16};B={8,4,16,18}It can be seen that 12ϵA but 12ϵB.∴A=B(iii) A={2,4,6,8,10}B={x:x is a positive even integer and x≤10}={2,4,6,8,10}∴A=B(iv) A={x:x is a multiple of 10}So, A={10,20,30,40....}B={10,15,20,25,30....}It can be seen that 15ϵB but 15ϵA.∴A=B

Question 10

Are the following pair of sets equal ? Give reasons.(i) A={2,3},B={x:x is solution of x2+5x+6=0}(ii) A={x:x is a letter in the word FOLLOW}B={y:y is a letter in the word WOLF}

Accepted Answer

(i) A={2,3}B={x:x is a solution of x2+5x+6=0}x2+5x+6=0 x(x+3)+2(x+3)=0(x+2)(x+3)=0x=−2 or x=−3∴B={−2,−3}∴A=B(ii) A={x:x is a letter in the word FOLLOW}={F,O,L,W}B={y:y is a letter in the word WOLF}={W,O,L,F}The order in which the elements of a set are listed is not significant.∴A=B

Question 11

From the sets given below, select equal set:A={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2}E={−1,1},F={0,a},G={−1,1},H={0,1}

Accepted Answer

A={2,4,8,12}n(A)=4B={1,2,3,4} n(B)=4C={4,8,12,14}n(C)=4D={3,1,4,2}n(D)=4E={−1,1}n(E)=2F={0,a}n(F)=2G={1,−1}n(G)=2H={0,1}n(H)=2Number of elements in A,B,C,D are same i.e. all have 4 elements. So ,they are comparable. Now, we see B and D has same elements.So, B and D are equal sets.Similarly, E,F,G, H are comparable as they all have same number of elements i.e. 2.Clearly , E and G has same elements.So, E and G are equal sets.

Question 12

Make correct statements by filling in the symbols ⊂ or ⊂ in blank spaces:(i) {2,3,4}......{1,2, 3,4,5} (ii) {a,b,c}...{b,c,d}(iii) {x:x is a student of Class XI of your school}. . .{x:x student of your school}(iv) {x:x is a circle in the plane} . . .{x:x is a circle in the same plane with  radius 1 unit}(v) {x:x is a triangle in a plane} . . . {x:x is a rectangle in the plane}(vi) {x:x is an equilateral triangle in a plane} . . . {x:x is a triangle in the same plane}(vii) {x:x is an even natural number} . . . {x:x is an integer}

Accepted Answer

(i) {2,3,4}⊂{1,2,3,4,5}(ii) {a,b,c}⊂{b,c,d}(iii) {x:x is a student of class XI of your school}⊂{x:x is student of your school}(iv) {x:x is a circle in the plane}⊂{x:x is a circle in the same plane with radius 1 unit}(v) {x:x is a triangle in a plane}⊂{x:x is a rectangle in the plane}(vi) {x:x is an equilateral triangle in a plane}⊂{x:x is a triangle in the same plane}(vii) {x:x is an even natural number}⊂{x:x is an integer}

Question 13

Examine whether the following statements are true or false:(i) { a,b } ⊂ { b,c,a }(ii) { a,e } ⊂ { x:x is a vowel in the English alphabet}(iii) { 1,2,3 } ⊂ { 1,3,5 }(iv) { a }⊂ { a,b,c }(v) { a }ϵ { a,b,c }(vi) { x:x is an even natural number less than 6} ⊂ { x : x is a natural number  which divides 36 }

Accepted Answer

(i) False. Each element of {a,b} is also an element of {b,c,a}.(ii) True. {a,e} are two vowels of the English alphabet.(iii) False. 2ϵ{1,2,3}However, 2∈/ {1,3,5}(iv) True. Each element of {a} is also an element of {a,b,c}.(v) False. The element of {a,b,c} are a,b,c. Therefore, a⊂a,b,c(vi) True. {x:xis an even natural number less than 6}={2,4}{x:xis a natural number which divides 36}={1,2,3,4,6,9,12,18,36}Here, {2,4}⊂ {1,2,3,4,6,9,12,18,36}

Question 14

Let A={1,2,{3,4},5}. Which of the following statements are incorrect and why?(i) {3,4}⊂A (ii) {3,4}ϵA (iii) {{3,4}}⊂A(iv) 1ϵA (v) 1⊂A (vi) {1,2,5}⊂A(vii) {1,2,5}ϵA (viii) {1,2,3}⊂A (ix) ϕ ϵA(x) ϕ⊂A (xi) {ϕ}⊂A

Accepted Answer

Given A={1,2,{3,4},5}(i) The statement {3,4}⊂A is incorrect because 3∈{3,4}; however, 3∈/A.(ii) The statement {3,4}∈A is correct because {3,4} is an element of A.(iii) The statement {{3,4}}⊂A is correct because {3,4}∈{{3,4}} and {3,4}∈A.(iv) The statement 1∈A is correct because 1 is an element of A.(v) The statement 1⊂A is incorrect because an element of a set can never be a subset of itself.(vi) The statement {1,2,5}⊂ A is correct because each element of {1,2,5} is also an element of A.(vii) The statement {1,2,5}∈A is incorrect because {1,2,5} is not an element of A.(viii) The statement {1,2,3}⊂A is incorrect because 3∈{1,2,3}; however, 3∈/A.(ix) The statement ϕ∈A is incorrect because ϕ is not an element of A.(x) The statement ϕ⊂A is correct because ϕ is a subset of every set.(xi) The statement {ϕ}⊂A is incorrect because ϕϵ{ϕ}; however, ϕ∈/A.

Question 15

Write down all the subsets of the following sets(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ϕ

Accepted Answer

We know that number of subsets of any set is 2n where n is the number of elements in that set.Every non-empty set has two subsets ϕ and the set itself.(i) The subsets of {a} are ϕ and {a}.(ii) The subsets of {a,b} are ϕ,{a},{b},and{a,b}.(iii) The subsets of {1,2,3} are ϕ,{1},{2},{3},{1,2},{2,3},{1,3},and{1,2,3}(iv) The only subset of ϕ is ϕ.

Question 16

How many elements has P(A) , if A=ϕ?

Accepted Answer

We know that if A is a set with m elements i.e., n(A)=m, then, n[P(A)]=2m.If A=ϕ, then n(A)=0∴n[P(A)]=2o=1Hence, P(A) has one element.

Question 17

Write the following as intervals:(i) {x:x∈R,−4<x≤6}(ii) {x:x∈R,−12<x<−10} (iii) {x:x∈R,0≤x<7} (iv) {x:x∈R,3≤x≤4}

Accepted Answer

(i) {x:x∈R,−4<x≤6}=(−4,6] (ii) {x:x∈R,−12<x<−10}=(−12,−10) (iii) {x:x∈R,0≤x<7}=[0,7) (iv) {x:x∈R,3≤x≤4}=[3,4]

Question 18

Write the following intervals in set-builder form:(i) (−3,0) (ii) [6,12] (iii) (6,12] (iv) [23,5)

Accepted Answer

(i) (−3,0)={x:xϵR,−3<x<0}(ii) [6,12]={x:xϵR,6≤x≤12}(iii) (6,12]={x:xϵR,6<x≤12}(iv) [−23,5)={x:xϵR,−23≤x<5}

Question 19

What universal set(s) would you propose for each of the following:(i) The set of right triangles. (ii) The set of isosceles triangles

Accepted Answer

(i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons.(ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

Question 20

Given the sets A={1, 3, 5}, B={2, 4, 6} and C={0, 2, 4, 6, 8}, which of the  following may be considered as universal set (s) for all the three sets A, B and Ci){0, 1, 2, 3, 4, 5, 6} ii)ϕiii){0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}iv){0, 1, 2, 3, 4, 5, 6, 7, 8}

Accepted Answer

(i) It can be seen that A⊂{0,1,2,3,4,5,6}B⊂{0,1,2,3,4,5,6}However, C⊂{0,1,2,3,4,5,6}Therefore, the set {0,1,2,3,4,5,6} cannot be the universal set for the sets A, B and C.(ii) A⊂ϕ,B⊂ϕ,C⊂ϕTherefore, ϕ cannot be the universal set for the sets A, B, and C.(iii) A⊂{0,1,2,3,4,5,6,7,8,9,10}B⊂ {0,1,2,3,4,5,6,7,8,9,10}C⊂ {0,1,2,3,4,5,6,7,8,9,10}Therefore, the set {0,1,2,3,4,5,6,7,8,9,10} is the universal set for the sets A, B and C.(iv) A⊂{1,2,3,4,5,6,7,8}B⊂ {1,2,3,4,5,6,7,8}However, C⊂{1,2,3,4,5,6,7,8}Therefore, the set {1,2,3,4,5,6,7,8} cannot be the universal set for the sets A, B, and C.

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