Power Set

Power Set

## Definition / Concept

The power set of $A$ is the set containing all subsets of $A$.

## Formula

$$P(A)=\{B:B\subseteq A\}$$

If:

$$n(A)=n$$

then:

$$n(P(A))=2^n$$

## Conditions / Special Cases

$$\varnothing\in P(A)$$

$$A\in P(A)$$

Number of proper subsets:

$$2^n-1$$

Number of non-empty subsets:

$$2^n-1$$

Number of non-empty proper subsets:

$$2^n-2$$

For the empty set:

$$P(\varnothing)=\{\varnothing\}$$

$$n(P(\varnothing))=1$$

## Usage

- Used in problems involving the number of subsets of a finite set.