Power Set

A power set includes all the subsets of a given set including the empty set. A power set can be imagined as a place holder of all the subsets of a given set, or, in other words, the subsets of a set are the members or elements of a power set.

Definition

A power set is defined as the set or group of all subsets for any given set, including the empty set, which is denoted by $\{\}$, or, $\varnothing$. A set that has $n$ elements has $2^{n}$ subsets in all.

For example,

Let Set $A=\{1,2,3\}$, number of elements in the set is $3$. Therefore, there are $2^{3}$elements in the power set.

Let us find the power set of set $A$.

Subsets of set $A=\{\},\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}$

Power set $P(A)=\{\},\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}$

The cardinality of a power set = $2^{n}$ = $2^{3}$= $8$