Problems & Solutions : Sets : Exercise 1(a) : Intermediate
Exercise 1(a) : ( problems & solutions )
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.
Solution :
Suppose A is the collection of months of a year beginning with the letter J are January, June and
July.
∴ A= { January, June,July }
This A is well-defined i.e. it is unchanged.
Hence A is a set.
(ii) The collection of ten most talented writers of India.
Solution :
The collection of ten most talented writers of India may vary from person to person.
Hence that collection is not a set.
(iii) A team of eleven best-cricket batsmen of the world.
Solution :
A team of eleven best - cricket batsmen of the world may vary from person to person.
Hence that collection is not a set.
(iv) The collection of all boys in your class .
Solution :
The collection of all boys in your class room is exactly known and is well-defined
Hence that collection is a set.
(v) The collection of all natural numbers less than 100.
Solution :
Let A be the collection of natural numbers less than 100.
∴ A = { 1,2,3,...,99}
The elements in A are well- known and hence A is well-defined.
Hence A is a set.
(vi) A collection of novels written by the writer Munshi Prem Chand.
Solution :
The collection of novels written by the writer Munshi prem Chand is well-known and
well- defined.
Hence that collection is a set.
( vii) the collection of all even integers.
Solution :
The collection of all even integers A = { ...-4,-2,0,2,4,6,...} is known and well-defined.
∴ the collection A is a set.
(viii) The collection of questions in a chapter .
Solution :
The collection of questions in a chapter is known and well-defined, the collection is a set.
(ix) A collection of most dangerous animals of the world.
Solution :
The collection of most dangerous animals of the world may vary from person to person , the
collection is a set.
2. 2. Let (A={1,2,3,4,5,6}). Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5 . . . A
Solution :
Since 5 is in the set A, 5 𝞊 A
(ii) 8 . . . A
Solution :
Since 8 is not in the set A, 8 ∉ A .
(iii) 0 . . . A
Solution :
Since 0 is not in the set A, 0 ∉ A
(iv) 4 . . . A
Solution :
Since 4 is in the set A, 4 𝞊 A
(v) 2 . . . A
Solution :
Since 2 is in the set A, 2 𝞊 A
(vi) 10 . . . A
3. Write the following sets in roster form:
(i) A = { x : x is an integer and -3 ≤ x < 7 }
Solution :
Given A = { x : x is an integer and -3 ≤ x < 7 } .
i.e the set A containing integers lies between -3 and 7 together with -3.
∴ A = { -3,-2,-1,0,1,2,3,4,5,6 }
(ii) B = { x : x is a natural number less than 6 }
Solution :
Given B = { x : x is a natural number less than 6 }
= { 1,2,3,4,5}
(iii) C = { x : x is a two-digit natural number such that the sum of its digits is 8 }
Solution :
Given C = { x : x is a two-digit natural number such that the sum of its digits is 8 }.
we have 8 = 0+8 = 8+0
= 1+7 = 7+1
= 2+6 = 6+2
= 3+5 = 5+3
= 4+4
The two-digit natural numbers whose sum of digits is 8 are 08,80,17,71,16,62,35,53,44.
∴ = { 8, 16,17,35,44,53,62,71,80}
(iv) D ={ x : x is a prime number which is divisor of 60 }
Solution :
Given D ={ x : x is a prime number which is divisor of 60 }
we have 60 = 1 x 60
= 2 x 30
= 3 x 20
= 4 x 15
= 5 x 12
= 6 x 10
Here the divisors of 60 are 1,2,3,4,5,6,10,12,15,20,30,60
Among these divisors , the prime numbers are 2,3,5 only.
∴ D = { 2,3,5 }
(v) E = The set of all letters of the word TRIGONOMETRY
Solution :
Given E = The set of all letters of the word TRIGONOMETRY
In the given word, there are 12 letters out of which the letters T, R and O are
repeated.
∴ E = { T,R,I,G,O,N,M,E,Y}
(vi) F = The set of all letters in the word BETTER.
Solution :
Given F = The set of all letters in the word BETTER.
In the given word, there are 6 letters out of which the letters E & T are repeated.
∴ F = { B,E,T,R}
4. Write the following sets in the set-builder form:
(i) {3,6,9,12}
Solution :
Given set = { 3,6,9,12 }
= { 3x1 , 3x2, 3x3 , 3x4 }
= { x : x= 3n, n 𝞊 N and 1 ≤ n ≤ 4 }
(ii) {2,4,8,16,32}
Solution :
Given set = { 2,4,8,16,32 }
= { 21,22, 23,24,25 }
= { x : x = 2n , n 𝞊 N and 1 ≤ n ≤ 5 }
(iii) {5,25,125,625}
Solution :
Given set = {5,25,125,625}
= { 51,52, 53,54 }
= { x : x = 5n , n 𝞊 N and 1 ≤ n ≤ 4 }
(iv) {2,4,6, ...}
Solution :
Given set = { 2,4,6,...}
= { 2x1, 2x2 , 2x3,...}
= { x : x = 2n , n 𝞊 N }
= The set of all even integers.
(v) {1,4,9,...,100}
Solution :
Given set = { 1,4,9,...,100}
= { 12,22,32,…,102}
= { x: x = n2, n 𝞊 N, 1≤n≤10 }
5. List all the elements of the following sets:
(i) A = { x : x is an odd natural number }
Solution :
We known that
Natural numbers = 1,2,3,4,5,6,7,8,9,10,...
Odd natural numbers = 1,3,5,7,9,...
∴ A = { 1,3,5,7,...}
(ii) B={x:x is an integer, -1/2 < x< 9/2}
Given B={ x: x is an integer, -1/2 < x< 9/2}
= { x : x is an integer, -0.5 < x < 4.5 }
= { 0,1,2,3,4 }
(iii) C={x:x is an integer,x2≤4}
Solution :
Given C={x:x is an integer,x2≤4}
We known that integers = ....,-4,-3,-2,-1,0,1,2,3,4,...
For x = ...,-4,-3,-2,-1,0,1,2,3,4,...
x2= ..., 16,9,4,1,0,1,4,9,16,...
since x2≤4,
For x = -3 , (-3)2 = 9
x = -2 , (-2)2 = 4
x = -1 , (-1)2 = 1
x = 0 , (0)2 = 0
x = 1 , (1)2 = 1
x = 2 , (2)2 = 4
x = 3 , (3)2 = 9 ...
Since squares of -2,-1,0,1 and 2 are less than or equal to 4,
C = { -2,-1,0,1,2 }
(iv) D = {x: x is a letter in the word “LOYAL”}
Solution :
Given D = {x: x is a letter in the word “LOYAL”}
= { L , O , Y , A } | Since L repeated twice |
(v) E={ x: x is a month of a year not having 31 days}
Solution :
Given E={ x: x is a month of a year not having 31 days}
The months February , April, June , September, November does not having 31 days.
∴ E = { Febrary, April, June, September , November }
(vi) F={ x: x is a consonant in the English alphabet which precedes k}
Solution :
Given F={ x: x is a consonant in the English alphabet which precedes k}
The English alphabet before k are a,b,c,d,e,f,g,h,i,j
Among them Vowels are a,e,i
∴ The consonents before k are b,c,d,f,g,h,j
Hence F = { b,c,d,f,g,h,j }
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