Given a set $S$, the power set of $S$ is the set $\mathcal{P}S$ of all subsets of $S$. Equivalently, it is
the set $\TV^S$ of all functions from $S$ to the set $\TV$ of truth values. This is often written $2^S$, since there are (at least in classical logic) exactly $2$ truth values;
the collection of subobjects of $X$ in the topos Set.
the slice category $Inj/S$, where Inj is the wide subcategory of Set with morphisms restricted to injections. This is similar to the subobject definition but is more unpacked. $Inj/S$ has objects that are injections to $S$ and morphisms that are commuting triangles of injections.
One generally needs a specific axiom in the foundations of mathematics to ensure the existence of power sets. In material set theory, this can be phrased as follows:
If $S$ is a set, then there exists a set $\mathcal{P}$ such that $A \in \mathcal{P}$ if $A \subseteq S$.
One can then use the axiom of separation (bounded separation is enough) to prove that $\mathcal{P}$ may be chosen so that the subsets of $A$ are the only members of $\mathcal{P}$; the axiom of extensionality proves that this $\mathcal{P}$ is unique.
In structural set theory, we state rather that there exists a set $\mathcal{P}$ which indexes the subsets of $A$ and prove uniqueness up to unique isomorphism.
In predicative mathematics, the existence of power sets (along with other βimpredicativeβ axioms) is not accepted. However we can still speak of a power set as a proper class, sometimes called a power class.
One can use power sets to construct function sets; the converse also works using excluded middle (or anything else that will guarantee the existence of the set of truth values). In particular, power sets exist in any theory containing excluded middle and function sets; thus predicative theories which include function sets must also be constructive.
The power set $\mathcal{P}S$ is a poset ordered by containment: $A$ precedes $B$ means that $A$ is a subset of $B$ ($A \subseteq B$).
Cantor's theorem states that there exists no surjection from $S$ to $\mathcal{P}S$; as there does exist such an injection, one concludes that
in the usual arithmetic of cardinal numbers.
Power sets live in the category Set. Given an object $S$ of any category, one can similarly form a poset of subobjects of $S$; the category is called well-powered when this poset is small. One also has an internal notion of power set (a power object) in a topos.
The power set construction constitutes an equivalence of categories between the opposite category Set$^{op}$ and that of complete atomic Boolean algebras. See at Set β Properties β Opposite category and Boolean algebras. Restricted to finite sets, the power set construction constitutes an equivalence of categories between the opposite category of FinSet and that of finite Boolean algebras. See at FinSet β Opposite category.