Sets

Question 1.
Which of the following are sets? Justify your answer.

• i) The collection of all the months of a year beginning with die 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 even integers.

Answer:

• i) There are 3 months as January, June and July beginning with letter ‘J’. Therefore, it is a well defined collection of months and hence it is a set.
• ii) The concept of talented writers of India is vague, since there is no rule given for deciding whether a particular writer is talented or not.
• Hence, the given collection is not a set.
• iii) A team of eleven best cricket batsmen of the world is vague, since there is no rule given for deciding whether a particular batsman is the best.
• Hence, the given collection is not a set.
• iv) The collection of all boys in my class is well defined. Hence, the given collection is a set.
• v) The collection of all even integers i.e., (2, 4, 6, 8, ……) is well defined.
• Hence, the given collection is a set.

Question 2.
If A = {0, 2, 4, 6}, B = {3, 5, 7} and C = {p, q, r} then fill the appropriate symbol, ∈ or ∉ in the blanks.

• i) 0 ….. A
• ii) 3 ….. C
• iii) 4 ….. B
• iv) 8 ….. A
• v) p ….. C
• vi) 7 ….. B

Answer:

• i) ∈
• ii) ∉
• iii) ∉
• iv) ∉
• v) ∈
• vi) ∈

Question 3.
Express the following statements using symbols.

• i) The elements ‘x’ does not belong to ‘A’.
• ii) ‘d’ is an element of the set ‘B’.
• iii) ‘1’ belongs to the set of Natural numbers N.
• iv) ‘8′ does not belong to the set of prime numbers P.

Answer:

• i) x ∉ A
• ii) d ∈ B
• iii) 1 ∈ N
• iv) 8 ∉ P

Question 4.
State whether the following statements are true or false. Justify your answer.

• i) 5 ∉ set of prime numbers
• ii) S = {5, 6, 7} implies 8 ∈ S.
• iii) -5 ∉ W where ‘W’ is the set of whole numbers.
• iv)8/11 ∈ Z

where ‘Z’ is the set of integers.

Answer:

• i) False
• ii) False
• iii) True
• iv) False

Question 5.
Write the following sets in roster form.

• i) B = {x : x is a natural number smaller than 6}.
• ii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}.
• iii) D = {x : x is a prime number which is a divisor of 60}.
• iv) E = {x : x is an alphabet in BETTER}.

Answer:

• i) B = {1, 2, 3, 4, 5}
• ii) C = {17, 26, 35, 44, 53, 62, 71}
• iii) D = {5, 3}
• iv) E = {B, E, T, R}

Question 6.
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) {1, 4, 9, 16, 25, …, 100}

Answer:

• i) A = {x : x is multiple of 3 and less than 13}
• ii) B = {x : x = 2^P, 0 < P < 6, P ∈ N}
• iii) C = {x : x = 5^P, 0 < P < 5, P ∈ N}
• iv) D = {x : x = P², 0 < P < 11, P ∈ W}

Question 7.
Write the following sets in roster form.

• i) A = {x: x is a natural number greater than 50 but smaller than 100}
• ii) B = {x : x is an integer, x² = 4}
• iii) D = {x : x is a letter in the word “LOYAL”}

Answer:

• i) A = {51, 52, 53, ……. , 98, 99}
• ii) B = {+2, -2}
• iii) D = {L, O, Y, A}

Question 8.
Match the roster form with set builder form.

i) {1, 2, 3, 6} ( ) a) {x : x is a prime number and a divisor of 6}
ii) {2, 3} ( ) b) {x : x is an odd natural number smaller than 10}
iii) {M, A, T, H, E, I, C, S} ( ) c) {x : x is a natural number and divisor of 6}
iv) {1, 3, 5, 7, 9} ( ) d) {x : x is a letter of the word MATHEMATICS}

Answer:

• i) c
• ii) a
• iii) d
• iv) b

Question 1.
If A = {1, 2, 3, 4}; B = {1, 2, 3, 5, 6} then find A ∩ B and B ∩ A. Are they equal ?

Answer:
Given sets are A = {1, 2, 3, 4} and B = {1,2,3, 5,6}
A ∩ B = {1,2, 3,4} ∩ {1,2, 3, 5, 6}
= {1,2,3} …… (1)
B ∩ A = {1, 2, 3, 5, 6} ∩ {1, 2, 3, 4}
= {1,2,3} …….(2)
From (1) and (2)
A ∩ B and B ∩ A are same.

Question 2.
A = {0, 2, 4}, find A ∩ ϕ and A ∩ A. Comment.

Answer:
Given set A = {0, 2, 4} and ϕ is a null set.
A ∩ ϕ = {0, 2, 4} ∩ { }
= { } ……. (1)
A ∩ A = {0, 2, 4} ∩ {0, 2, 4}
= {0, 2,4} …….. (2)
From (1) and (2),
We conclude that A ∩ ϕ = ϕ and A ∩ A = A

Question 3.
If A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}, find A - B and B - A.

Answer:
Given sets are
A {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}
A - B = {2, 4, 6, 8, 10} - {3, 6, 9, 12, 15}
= {2, 4, 8, 10} …… (1)
B – A = {3, 6, 9, 12, 15} - {2, 4, 6, 8, 10}
= {3, 9, 12, 15} …… (2)
From (1) and (2), A - B ≠ B - A

Question 4.
If A and B are two sets such that A ⊂ B then, what is A ∪ B?

Answer:
Let us consider A ⊂ B
Set A = {1, 2, 3} and
Set B = {1, 2, 3, 4, 5}
Now A ∪ B = {1, 2, 3} ∪ {1, 2, 3, 4, 5}
= {1, 2, 3, 4, 5} = B
∴ A ∪ B = B

Question 5.
If A = {x : x is a natural number},
B = {x : x is an even natural number},
C = {x : x is an odd natural number} and
D = {x : x is a prime number}
Find A ∩ B, A ∩ C, A ∩ D, B ∩ C, B ∩ D, C ∩ D.

Answer:
Given sets are
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ……}
B = {2, 4, 6, 8, 10, …….}
C = {1, 3, 5, 7, 9, …….}
D = {2, 3, 5, 7, 11, …….}
A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …….} ∩ {2, 4, 6, 8, 10, ……}
= {2, 4, 6, 8, 10, ……}
A ∩ C = {1, 2, 3,4, 5, 6, 7, 8, 9, 10, …} ∩ {1, 3, 5, 7, 9 }
= {1, 3, 5, 7, 9, ……}
A ∩ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} ∩ {2, 3, -5, 7, 11,….}
= {2, 3, 5, 7, 11, ……}
B ∩ C = {2, 4, 6, 8, 10, ……} ∩ {1, 3, 5, 7, 9, …….}
= { } = ϕ
B ∩ D = {2, 4, 6, 8, 10, ……} ∩ {2, 3, 5, 7, 11, ……}
= {2}
C ∩ D = {1, 3, 5, 7, 9, ……} ∩ {2, 3, 5, 7, 11, 13, ……}
= {3, 5, 7, …..}

Question 6.
If A = {3, 6, 9, 12, 15, 18, 21}; B = {4, 8, 12, 16, 20}; C = {2, 4, 6, 8, 10, 12, 14, 16}; D = {5, 10, 15, 20} find

• (i) A - B
• (ii) A - C
• (iii) A - D
• (iv) B - A
• (v) C - A
• (vi) D - A
• (vii) B - C
• (viii) B - D
• (ix) C - B
• (x) D - B

Answer:
Given sets are A = {3, 6, 9, 12, 15, 18, 21}
B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16} and
D = {5, 10, 15, 20}

• i) A - B = {3, 6, 9, 12, 15, 18, 21} - {4, 8, 12, 16, 20} = {3, 6, 9, 15, 18, 21}
• ii) A - C = {3, 6, 9, 12, 15, 18, 21} - {2, 4, 6, 8, 10, 12, 14, 16} = {3,9,15,18,21}
• iii) A - D = {3, 6, 9, 12, 15, 18, 21} - {5, 10, 15, 20} = {3,6,9,12,18,21}
• iv) B - A = {4, 8, 12, 16, 20} - {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}
• v) C - A = {2, 4, 6, 8, 10, 12, 14, 16} - {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
• vi) D - A = {5, 10, 15, 20} - {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
• vii) B - C = {4, 8, 12, 16, 20} - {2, 4, 6, 8, 10, 12, 14, 16} = {20}
• viii) B - D = {4, 8, 12, 16,20} - {5, 10, 15, 20} = {4, 8, 12, 16}
• ix) C - B = {2, 4, 6, 8, 10, 12, 14, 16} - {4, 8, 12, 16, 20} = {2, 6, 10, 14}
• x) D - B = {5, 10, 15, 20} - {4, 8, 12, 16, 20} = {5, 10, 15}

Question 7.
State whether each of the following statement is true or false. Justify your answers.

• i) {2,3,4,5} and {3,6} are disjoint sets.
• ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
• iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
• iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Answer:

i) Rule: If two sets are disjoint their intersection is null set.
= {2, 3, 4, 5} n {3, 6} = { 3 } ≠ ϕ
∴ Given statement is False.

ii) Given sets are
{a, e, i, o, u} and {a, b, c, d}
= {a, e, i, o, u} ∩ {a, b, c, d}
= { a } ≠ ϕ
∴ Given statement is False.

iii) Given sets are
{2, 6, 10, 14} and {3, 7, 11, 15}
= {2, 6, 10, 14} ∩ {3, 7, 11, 15}
= { }
∴ Given statement is True.

iv) Given sets are
{2, 6, 10} and {3, 7, 11}
= {2, 6, 10} ∩ {3, 7, 11} = { }
∴ Given statement is True.

Question 1.
Which of the following sets are equal?

• i)A = {x : x is a letter in the word FOLLOW}
• ii) B = {x : x is a letter in the word FLOW}
• iii) C = {x : x is a letter in the word WOLF}

Answer:

• i) Elements in set A are {F, L, O, W}
• ii) Elements in set B are {F, L, O, W}
• iii) Elements in set C are {F, L, O, W} Sets A, B and C have same elements, Hence, they are equal sets.

Question 2.
Consider the following sets and fill up the blank in the statement given below with = or ≠ so as to make the statement true.

A = {1, 2, 3};
B = {The first three natural numbers};
C = {a, b, c, d};
D = {d, c, a, b};
E = {a, e, i, o, u};
F = {Set of vowels in English Alphabet}

• i) A …. B
• ii) A …. E
• iii) C …. D
• iv) D …. F
• v) F …. A
• vi) D …. E
• vii) F …. B

Answer:

• i) A = B
• ii) A ≠ E
• iii) C = D
• iv) D ≠ F
• v) F ≠ A
• vi) D ≠ E
• vii) F ≠ B

Question 3.
In each of the following, state whether A = B or not.

• i) A = {a, b, c, d} ; B = {d, c, a, b}
• 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, …}

Answer:

• i) A = B
• ii) A ≠ B
• iii) A ≠ B
• iv) A ≠ B

Question 4.
State the reasons for the following :

i) {1, 2, 3, …., 10} ≠ {x : x ∈ N and 1 < x < 10}
ii) {2, 4, 6, 8, 10} ≠ {x : x = 2n+1 and x ∈ N}
iii) {5, 15, 30, 45} ≠ {x : x is a multiple of 15}
iv) {2, 3, 5, 7, 9} ≠ {x : x is a prime number}

Answer:

i) In R.H.S ‘x’ is greater than 1 and less than 10 but L.H.S is having both 1 and 10.
ii) L.H.S ≠ R.H.S
R.H.S: x = 2n + 1 is definition of odd numbers.
L.H.S: Given set is even numbers set.
iii) x is a multiple of 15.
So 5 does not exist.
iv) x is a prime number but 9 is not a prime number.

Question 5.
List all the subsets of the following sets.

• i) B = {p, q}
• ii) C = {x, y, z}
• iii) D = {a, b, c, d}
• iv) E = {1, 4, 9, 16}
• v) F = {10, 100, 1000}

Answer:

• i) Subsets of ‘B’ are {p}, {q}, {p, q}, ϕ
• ii) Subsets of ‘C’ are {x}, {y} {z}, {x, y}, {y, z}, {z, x}, {x, y, z} and ϕ (2³= 8)
• iii) Subsets of ‘D’ are {a}, {b}, {c}, {d}, {a,b}, {b,c}, {c, d}, {a, c}, {a, d}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, d}, {a, b, c, d} and φ
• iv) Subsets of ‘E’ are
• ϕ, {1}, {4}, {9}, {16}, {1,4}, {1,9}, {1, 16}, {4, 9}, {4, 16}, (9, 16}, {1, 4, 9}, {1, 9, 16}, {4, 9, 16}, {1, 4, 16}, {1, 4, 9, 16}
• v) Subsets of ‘F’ are
• ϕ, {10}, {100}, {1000}, {10, 100}, {100, 1000}, {10, 1000}, {10, 100, 1000}.

Question 1.
State which of the following sets are empty and which are not?

• i) The set of lines passing through a given point.
• ii) Set of odd natural numbers divisible by 2.
• iii) {x : x is a natural number, x < 5 and x > 7}
• iv) {x: x is a common point to any two parallel lines}
• v) Set of even prime numbers.

Answer:

• i) Not empty
• ii) Empty
• iii) Empty
• iv) Empty
• v) Not empty

Question 2.
Which of the following sets are finite or infinite?

• i) The set of months in a year.
• ii) {1, 2, 3, …, 99, 100}
• iii) The set of prime numbers smaller than 99.

Answer:

• i) Finite
• ii) Finite
• iii) Finite

Question 3.
State whether each of the following sets is finite or infinite.

• i) The set of letters in the English alphabet.
• ii) The set of lines which are parallel to the X-axis.
• iii) The set of numbers which are multiples of 5.
• iv) The set of circles passing through the origin (0, 0).

Answer:

• i) Finite
• ii) Infinite
• iii) Infinite
• iv) Infinite