---
tag: 0BC7KX51Z8
kind: definition
name: Power set
msc: ["03E20"]
defines: 00E0YZCERH
uses: [0FE8Q5S42Y]
sources:
  - url: https://en.wikipedia.org/wiki/Power_set
    locator: lead section
---

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The power set of $A$ is the set
$$\mathcal{P}(A) = \{S \mid S \subseteq A\}$$
of all [[subsets|0FE8Q5S42Y]] of $A$.

## Remarks

Other notations are $2^A$ and $\wp(A)$.
Example: $\mathcal{P}(\{1, 2\}) = \{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$.
A finite set with $n$ elements has a power set with $2^n$ elements. See [[cardinality of the power set|0GPFKS96T7]].
By [[Cantor's theorem|0WTEMGYV2C]], no set is in bijection with its power set.

## Explanations

### L1 human

To list the power set, choose for each element whether it is in or out of the subset.
Each choice gives one subset, so a set with $3$ elements has $2^3 = 8$ subsets.
The empty set and the set itself are always among them.