**How to Write Power Set for the Given Sets ?**

Here we are going to see how to write the power set for the given sets.

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)] = 2^{m}

(ii) The number of proper subsets of a set A is

n [P(A)] – 1 = 2^{m}–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 = 2^{m}–1

n(A) = 3

n [P(A)] – 1 = 2^{3}–1

= 8 - 1

**Number of proper subset :**

n [P(A)] – 1 = 7

(ii) X = { x^{2} : x ∈ N, x^{2} ≤ 100}

X = {2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, 7^{2}, 8^{2}, 9^{2}, 10^{2}}

n (X) = 10

n [P(A)] – 1 = 2^{10 }– 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)] = 2^{4 }

= 16

(ii) If n(A) = 0, find n [P(A)].

**Solution :**

n [P(A)] = 2^{m}

= 2^{0}

= 1

(iii) If n[P(A)] = 256, find n(A).

n [P(A)] = 256

2^{m }= 256

2^{m }= 2^{8}

m = 8

After having gone through the stuff given above, we hope that the students would have understood, "How to Write Power Set for the Given Sets".

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