# Power Set

## Power Set

Let A be a non-empty set, then collection of all possible subsets of a set A is known as Power Set. It is denoted by P(A).

Example: Suppose A = {1, 2, 3}

P(A) = {Ф, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}}

• Ï P(A)
• [A] ϵ P(A)

Properties of power set:

(i) Each element of a power set is a set.

(ii)If A Í B, then P(A) Í P(B)

(iii)Power set of any set is always non empty

(iv)If a set A has n elements, then P(A) has 2ⁿ elements

(v) P (A)  È P (B) Í P (A È B)

Examples 1: If set A = {1, 3, 5}, then number of elements in P{P(A}} is

Solution:Given that A = {1, 3, 5}

If a set A has n elements, then P(A) has 2ⁿ elements

n{P(A)} = 2ⁿ

where n = 3

n{P(A)} = 2³

= 2 x 2 x 2 = 8

n{P(P(A))} = 2n

where n = 8

n{P(A)} = 2⁸

= 256

Example 2: If set A = {1, 2, 3, 4, 5, 6}, then number of elements in P{P(A}} is

Solution: Given that A = {1, 2, 3, 4, 5, 6}

If a set A has n elements, then P(A) has 2ⁿ elements

n{P(A)} = 2ⁿ

where n = 6

n{P(A)} = 2⁶

n{P(P(A))} = 2n

where n = 64

n{P(A)} = 2⁶⁴.