---
> [!definition]+ Definition. ([[power set]])
> The **power set** of a set $S$, denoted $\mathcal{P}(S)$ or $2^{S}$ is the set of all subsets of $S$, including [[the empty set]] and $S$ itself.
^definition
> [!basicproperties]+
> - If $S$ is finite then $|\mathcal{P}(S)|=2^{|S|}$.
> - The power set of a [[countably infinite]] set is [[uncountably infinite]].
^properties
> [!proof]+
> - For each subset $A \subset S$, define an indicator function $\mathbb{1}_{A}: S \to \{ 0,1 \}$ to be given by $\mathbb{1}_{A}(s)=\begin{cases}1 & s \in A; \\ 0 & \text{else}.\end{cases}$ Noting that every function $f:S \to \{ 0,1 \}$ is the indicator for a unique $A \subset S$ (namely, the set $\{ s \in S: f(s)=1 \}$), we see that the task of counting how many subset of $S$ exist amounts to counting how many functions from $S$ to $\{ 0,1 \}$ exist. This is $2^{|S|}$, by the [[multiplication principle]].
^proof
---
#### References
> [!backlink]-
> ```dataview
TABLE rows.file.link as "Further Reading"
FROM [[]]
FLATTEN file.tags
GROUP BY file.tags as Tag
> [!frontlink]-
> ```dataview
TABLE rows.file.link as "Further Reading"
FROM outgoing([[]])
FLATTEN file.tags
GROUP BY file.tags as Tag
#reformatrevisebatch02