Which of the following collections are sets? If any collection is not a set, give reasons.
(i) All persons on earth
(ii) All male teachers of your school
(iii) All leaders of India
(iv) All healthy people of your town
(v) All fat boys of your locality
(vi) All planets in our solar system
(vii) All difficult problems in your book on Mathematics
(viii) All 3-digit natural numbers
(ix) All books in your school library
Answer
(i) It is a set as it is well defined. For any given person, we can definitely tell whether that person is on earth or not.
(ii) It is a set as it is well defined. The male teachers of your school can be clearly identified.
(iii) It is not a set because the given collection is not well defined—people may differ on whether a particular person is a leader of India or not.
(iv) It is not a set because the given collection is not well defined—people may differ on whether a person of the town is healthy or not.
(v) It is not a set because the given collection is not well defined—people may differ on whether a boy of the locality is fat or not.
(vi) It is a set as it is well defined. The planets in our solar system can be clearly listed.
(vii) It is not a set because the given collection is not well defined—people may differ on whether a problem is difficult or not.
(viii) It is a set as it is well defined. The 3-digit natural numbers are 100, 101, 102, ......, 999.
(ix) It is a set as it is well defined. The books in your school library can be clearly identified.
Describe the following set in Roster form:
A = {x : x is a natural number, 5 < x < 14}
Answer
The natural numbers x such that 5 < x < 14 are 6, 7, 8, 9, 10, 11, 12 and 13.
Hence, A = {6, 7, 8, 9, 10, 11, 12, 13}.
Describe the following set in Roster form:
B = {x : x is a factor of 18}
Answer
The factors of 18 are 1, 2, 3, 6, 9 and 18.
Hence, B = {1, 2, 3, 6, 9, 18}.
Describe the following set in Roster form:
C = {x : x is a prime factor of 330}
Answer
330 = 2 × 3 × 5 × 11
So, the prime factors of 330 are 2, 3, 5 and 11.
Hence, C = {2, 3, 5, 11}.
Describe the following set in Roster form:
D = {x : x is a 2-digit number, the sum of whose digits is 8}
Answer
The 2-digit numbers whose digits add up to 8 are 17, 26, 35, 44, 53, 62, 71 and 80.
Hence, D = {17, 26, 35, 44, 53, 62, 71, 80}.
Describe the following set in Roster form:
E = {x : x is a natural number, x is a multiple of 7, x < 60}
Answer
The multiples of 7 that are less than 60 are 7, 14, 21, 28, 35, 42, 49 and 56.
Hence, E = {7, 14, 21, 28, 35, 42, 49, 56}.
Describe the following set in Roster form:
F = {x : x is a natural number, x is neither prime nor composite}
Answer
The only natural number that is neither prime nor composite is 1.
Hence, F = {1}.
Describe the following set in Roster form:
G = {x : x is an even prime number}
Answer
The only even prime number is 2.
Hence, G = {2}.
Describe the following set in Roster form:
H = {x : x is an odd natural number, 50 < x < 60}
Answer
The odd natural numbers x such that 50 < x < 60 are 51, 53, 55, 57 and 59.
Hence, H = {51, 53, 55, 57, 59}.
Describe the following set in set-builder form:
A = {16, 18, 20, 22, 24, 26, 28}
Answer
The elements 16, 18, 20, 22, 24, 26, 28 are even natural numbers lying between 14 and 30.
Hence, A = {x : x is an even natural number, 14 < x < 30}.
Describe the following set in set-builder form:
B = {23, 25, 27, 29, 31, 33, 35}
Answer
The elements 23, 25, 27, 29, 31, 33, 35 are odd natural numbers lying between 22 and 36.
Hence, B = {x : x is an odd natural number, 22 < x < 36}.
Describe the following set in set-builder form:
C = {1, 3, 5, 9, 15, 45}
Answer
The elements 1, 3, 5, 9, 15, 45 are the factors of 45.
Hence, C = {x : x is a natural number, x is a factor of 45}.
Describe the following set in set-builder form:
D = {15, 24, 33, 42, 51, 60}
Answer
The elements 15, 24, 33, 42, 51, 60 are 2-digit numbers, the sum of whose digits is 6.
Hence, D = {x : x is a 2-digit number, the sum of whose digits is 6}.
Describe the following set in set-builder form:
E = {6, 12, 18, 24, 30, 36, 42, 48}
Answer
The elements 6, 12, 18, 24, 30, 36, 42, 48 are multiples of 6 that are less than 50.
Hence, E = {x : x is a natural number, x is a multiple of 6, x < 50}.
Describe the following set in set-builder form:
F = {41, 43, 47, 53, 59, 61, 67, 71, 73}
Answer
The elements 41, 43, 47, 53, 59, 61, 67, 71, 73 are prime numbers lying between 40 and 74.
Hence, F = {x : x is a prime number, 40 < x < 74}.
Describe the following set in set-builder form:
G = {0, 1, 4, 9, 16, 25, 36, 49}
Answer
The elements 0, 1, 4, 9, 16, 25, 36, 49 are the squares of the whole numbers 0, 1, 2, 3, 4, 5, 6 and 7.
So, each element is of the form n2, where n is a whole number and 0 ≤ n ≤ 7.
Hence, G = {x : x = n2, where n is a whole number, 0 ≤ n ≤ 7}.
Let A = {5, 7, 11, 13, 17, 19, 23, 29}.
Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 7 ......... A
(ii) 9 ......... A
(iii) 3 ......... A
(iv) 15 ......... A
(v) 17 ......... A
(vi) 25 ......... A
(vii) 11 ......... A
(viii) 31 ......... A
Answer
(i) 7 is a member of A.
Hence, 7 ∈ A.
(ii) 9 is not a member of A.
Hence, 9 ∉ A.
(iii) 3 is not a member of A.
Hence, 3 ∉ A.
(iv) 15 is not a member of A.
Hence, 15 ∉ A.
(v) 17 is a member of A.
Hence, 17 ∈ A.
(vi) 25 is not a member of A.
Hence, 25 ∉ A.
(vii) 11 is a member of A.
Hence, 11 ∈ A.
(viii) 31 is not a member of A.
Hence, 31 ∉ A.
Let B = {2, 4, 6, 8, 10, ......, 74, 76, 78, 80}.
Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 26 ......... B
(ii) 31 ......... B
(iii) 70 ......... B
(iv) 15 ......... B
(v) 60 ......... B
(vi) 53 ......... B
(vii) 47 ......... B
(viii) 34 ......... B
Answer
B is the set of all even numbers from 2 to 80.
(i) 26 is an even number lying between 2 and 80.
Hence, 26 ∈ B.
(ii) 31 is an odd number.
Hence, 31 ∉ B.
(iii) 70 is an even number lying between 2 and 80.
Hence, 70 ∈ B.
(iv) 15 is an odd number.
Hence, 15 ∉ B.
(v) 60 is an even number lying between 2 and 80.
Hence, 60 ∈ B.
(vi) 53 is an odd number.
Hence, 53 ∉ B.
(vii) 47 is an odd number.
Hence, 47 ∉ B.
(viii) 34 is an even number lying between 2 and 80.
Hence, 34 ∈ B.
State whether the given statement is true or false:
(i) 0 ∈ {x : x is a whole number, x < 10}
(ii) -5 ∈ {x : x is an integer, -4 < x < 4}
(iii) -2 ∈ {x : x is an integer, x2 = 4}
(iv) 2 ∈ {x : x is a prime number, 3 < x < 6}
(v) 83 ∈ {x : x is a prime number, x < 100}
Answer
(i) The whole numbers less than 10 are 0, 1, 2, ......, 9. Since 0 is one of them, 0 is a member of this set.
True
(ii) The integers x such that -4 < x < 4 are -3, -2, -1, 0, 1, 2, 3. Since -5 is not one of them, -5 is not a member of this set.
False
(iii) x2 = 4 ⇒ x = 2 or x = -2. So the set is {-2, 2}. Since -2 is one of them, -2 is a member of this set.
True
(iv) The prime numbers x such that 3 < x < 6 give only x = 5. So the set is {5}. Since 2 is not a member of this set, the given statement is false.
False
(v) 83 is a prime number and 83 < 100, so 83 is a member of this set.
True