The set of all subsets of a set A is called the power set of ‘A’. It is denoted by P(A).
(i) If n(A) = m, then n[P(A)] = 2m.
(ii) The number of proper subsets of a set A is
n [P(A)] – 1 = 2m – 1
Question 1 :
Write down the power set of the following sets.
(i) A = {a, b}
Solution :
Subset of A are
= { { }, {a}, {b}, {a, b} }
(ii) B = {1, 2, 3}
Solution :
Subset of B are
= { { }, {1}, {2}, {3}, {1, 2}, {2, 3} {3, 1} {1, 2, 3} }
(iii) D = {p, q, r, s}
Solution :
Subset of D are
= {{ }, {p}, {q}, {r}, {s}, {p, q}, {p, r}, {p, s}, {q, r}, {q, s} {r, s}, {q, r, s} {p, q, r, s}}
(iv) E = ∅
Solution :
P(E) = { { } }
Question 2 :
Find the number of subsets and the number of proper subsets of the following sets.
(i) W = {red, blue, yellow}
Solution :
The number of proper subsets of a set A is
n [P(A)] – 1 = 2m–1
n(A) = 3
n [P(A)] – 1 = 23–1
= 8 - 1
Number of proper subset :
n [P(A)] – 1 = 7
(ii) X = { x2 : x ∈ N, x2 ≤ 100}
X = {22, 32, 42, 52, 62, 72, 82, 92, 102}
n (X) = 10
n [P(A)] – 1 = 210 – 1
n [P(A)] = 1024
Number of proper subset :
Number of proper subset = 1024 - 1 = 1023
Question 3 :
(i) If n(A) = 4, find n [P(A)].
Solution :
n(A) = 4, find n [P(A)]
n [P(A)] = 24
= 16
(ii) If n(A) = 0, find n [P(A)].
Solution :
n [P(A)] = 2m
= 20
= 1
(iii) If n[P(A)] = 256, find n(A).
n [P(A)] = 256
2m = 256
2m = 28
m = 8
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