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.