OP Malhotra Class 11 Maths Solutions Chapter 1 Sets Ex 1(b)

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S Chand Class 11 ICSE Maths Solutions Chapter 1 Sets Ex 1(b)

Question 1.
Find the subsets of
(i) (a)
(ii) {Reena, Sonu}
(iii) Φ
(iv) {5, {7}}
Solution:
(i) Required subsets are ; Φ, {a}
(ii) Required subsets are ; Φ, {Sonu}, {Reena}, {Sonu, Reena}
(iii) Required subsets are ; Φ
(iv) Required subsets are ; Φ, {5}, {{7}}, {5, {7}}

Question 2.
Let A = {p, q, r}
(i) List all the subsets of A.
(ii) List all the proper subsets of A.
Solution:
Given A = {p, q, r}
(i) Φ, {P}, {q}, {r}, {p, q}, {q, r), {p, r}, {p, q, r)
(ii) Φ, {p}, {q}, {r}, {p, q}, {q, r}, {p, r}

Question 3.
Let P = {whole numbers less than 30}
(i) List the subsets Q {even numbers}
(ii) List the subset R {odd numbers}
(iii) List the subset S {prime numbers}
(iv) List the subset T {square numbers}
(v) List the subset U {triangle numbers}
Solution:
Given P = {0, 1, 2, 3 29}
(i) Q = {0, 2, 4, 6, , 28}
(ii) R = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
(iii) S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
(iv) T = {0, 1,4,9, 16,25}
(v) U = {0, 1, 3, 6, 10, 15, 21, 28}

Question 4.
Tell in each of the following, whether first set is a subset of the second set or not.
(i) A = Set of letters in the word ‘LATE’
B = Set of letters in the word ‘PLATE’
(ii) P = Set of even prime numbers.
Q = {x | x = 2p, p ∈ N and 1 ≤ p ≤ 3}
(iii) L = Set of digits in the number 1590
M = Set of digits in the number 178902
(iv) E = Set of all triangles having 4 sides.
F = Set of digits in the number ‘100’.
Solution:
(i) A = {L, A, T, E} and B = {P, L, A, T, E}
Since every element of set A is a member of set B.
∴ A be a subset of B.

(ii) Given P = {2} since 2 be the only even prime number
and Q = {x | x = 2p, p ∈ N and 1 ≤ p ≤ 3}
Since p ∈ N and 1 ≤ p ≤ 3 ∴ p = { 1, 2, 3}
When p = 1 ⇒ x = 2 x 1 = 2
When p = 2 ⇒ x = 2 x 2 = 4
When p = 3 ⇒ x = 2 x 3 = 6
∴ Q = {2, 4, 6}
Since every member of set A is a member of set B.
∴ A is a subset of B.

(iii) L = {0, 1,5,9} and M = {0, 1, 2, 7, 8, 9}
since 5 ∈ L but 5 ∉ M.
∴ L is not a subset of M.

(iv) E = set of all triangles having 4 sides since their is no triangle having four sides
∴ E = Φ and F = {0, 1}
Clearly E is a subset of F as empty set is a subset of every set.

Question 5.
Write the proper subsets of the following sets :
(i) {7}
(ii) {1, 3}
(iii) {c, a, b}
(iv) Φ
Solution:
(i) Φ
(ii) Φ, {1}, {3}
(iii) Φ, {c}, {a}, {b}, {c, a}, {a, b}, {c, b}
(iv) it has no proper subset

Question 6.
How many subsets do the following sets have ?
(i) A set having 5 elements.
(ii) The set of letters of the word ‘CENTENARY’
Solution:
(i) We know that a set having n elements has 2ⁿ subsets.
given set having 5 elements the no. of subsets of given set be 2⁵ i.e. 32.

(ii) Given set = {C, E, N, T, A, R, Y}
Clearly no. of elements in given set be 7.
∴ no. of subsets of given set = 27 = 128

Question 7.
How many proper subsets do the following sets have ?
(i) The set of factors of 12.
(ii) The set A {x | x is a prime numbers, x < 20}
Solution:
We know that if a set A has n elements. Then number of proper subsets of A are 2ⁿ – 1.
(i) A = set of all factors of 12 = {1, 2, 3, 4, 6, 12}
∴ no. of proper subsets of A = 2ⁿ – 1
= 2⁶ – 1 = 63

(ii) A = {2, 3, 5, 7, 11, 13, 17, 19}
Here n = 8
∴ number of proper subsets of A = 2⁸ – 1
= 255

Question 8.
Answer true or false
(i) 3 ⊆ {3, 0}
(ii) {3} ⊆ {3, 0}
(iii) Φ ⊂ {3, 0}
(iv) 0 ∈ {3, 0}
(v) Φ ⊂ {3, 0}
(vi) Φ ⊆ {Φ}
(vii) For any two sets A and B either A ⊆ B or B ⊆ A.
(viii) Every set has a proper subset.
(ix) Every subset of a finite set is finite.
(x) Every subset of an infinite set is infinite.
Solution:
(i) 3 ∈ {3, 0} but 3 is not a subset of {3, 0}
∴ given statement is false.

(ii) since {3} is a subset of {3, 0}
∴ given statement is true.

(iii) since φ is not a member of {3, 0}
∴ φ ∉ {3, 0}
Hence given statement is false.

(iv) Since 0 be a member of {3, 0}
∴ 0 ∈ {3, 0}
Thus given statement is true.

(v) Since φ i.e. empty set is proper subset of {3, 0}.
∴ φ ⊂ {3, 0}
Thus given statement is true.

(vi) empty set is a subset of every set.
∴ given statement is true.

(vii) False, A = {1, 2} and B = {4, 5}
A ⊄ B orB ⊄ A

(viii) Since empty set has no proper subset
∴ given statement is false.

(ix) True statement

(x) False, since empty set is a subset of every set and φ is a finite set.

Question 9.
Find the power set of each of the following sets :
(i) A = {digits in the number 98}
(ii) B = {letters in the word ‘KID’}
(iii) S = {2, 3}
(iv) T = {4, 7, 9}
Solution:
We know that, power set is the set of all subsets of given set.
(i) Given A = {8, 9}
∴ P (A) = {φ, {8}, {9}, {8, 9}}

(ii) B = {K, I, D}
∴ P (A) = {φ, {K}, {I}, {D}, {K, I}, {K, D}, {I, D}, {K, I, D}}

(iii) S = {2, 3}
∴ P (S) = {Φ, {2}, {3}, {2, 3}}

(iv) T = {4, 7, 9}
∴ P(T) = {Φ, {4}, {7}, {9}, {4, 7}, {4, 9}, {7, 9}, {4, 7, 9}}