Power Set — Definition, Formula & Examples

The power set of a set

S

is the set of all possible subsets of

S

, including the empty set and

S

itself.

Given a set $S$ , the power set $\mathcal{P}(S)$ is defined as $\mathcal{P}(S) = \{A : A \subseteq S\}$ . If $S$ has $n$ elements, then $\mathcal{P}(S)$ has exactly $2^n$ elements.

Key Formula

|\mathcal{P}(S)| = 2^n

Where:

$\mathcal{P}(S)$ = The power set of S
$n$ = The number of elements in S
$2^n$ = The number of subsets of S

How It Works

To build the power set of a set

S

, list every subset you can form by choosing to include or exclude each element. For each element, you have two choices — in or out — which is why the total count is

2^n

. Always remember to include the empty set

\emptyset

(which contains no elements) and

S

itself (which contains all of them). The result is a set whose elements are themselves sets.

Worked Example

Problem: Find the power set of

S = \{1, 2, 3\}

Step 1: Count the elements.

S

has

n = 3

elements, so the power set will have

2^3 = 8

subsets.

2^3 = 8

Step 2: List all subsets by systematically choosing to include or exclude each element.

\mathcal{P}(S) = \{\emptyset,\; \{1\},\; \{2\},\; \{3\},\; \{1,2\},\; \{1,3\},\; \{2,3\},\; \{1,2,3\}\}

Answer:

\mathcal{P}(S)

contains 8 subsets:

\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}

Why It Matters

Power sets appear in combinatorics whenever you need to count all possible selections from a group. In computer science, they model all possible states of a collection of binary flags or features. Discrete mathematics courses rely on power sets to prove properties about set operations and cardinality.

Common Mistakes

Mistake: Forgetting to include the empty set as a subset.

Correction: The empty set is a subset of every set, so it always belongs in the power set. A power set of a set with

n

elements must have exactly

2^n

members — if yours has fewer, you likely missed

\emptyset

or another subset.