This fact as well as the reason of the notation 2S denoting the power set P(S) are demonstrated in the below.

The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.

In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set.

For infinite Boolean algebras, this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem).

It can hence be shown, by proving the distributive laws, that the power set considered together with both of these operations forms a Boolean ring.

As "2" can be defined as {0, 1} (see, for example, von Neumann ordinals), 2S (i.e., {0, 1}S) is the set of all functions from S to {0, 1}.

As shown above, 2S and the power set of S, P(S), are considered identical set-theoretically.

This equivalence can be applied to the example above, in which S = {x, y, z}, to get the isomorphism with the binary representations of numbers from 0 to 2n − 1, with n being the number of elements in the set S or |S| = n. First, the enumerated set { (x, 1), (y, 2), (z, 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as {x, y} = 011(2); x of S is located at the first from the right of this sequence and y is at the second from the right, and 1 in the sequence means the element of S corresponding to the position of it in the sequence exists in the subset of S for the sequence while 0 means it does not.

For the whole power set of S, we get: Such an injective mapping from P(S) to integers is arbitrary, so this representation of all the subsets of S is not unique, but the sort order of the enumerated set does not change its cardinality.

(E.g., { (y, 1), (z, 2), (x, 3) } can be used to construct another injective mapping from P(S) to the integers without changing the number of one-to-one correspondences.)

(In this example, x, y, and z are enumerated with 1, 2, and 3 respectively as the position of binary digit sequences.)

The binomial theorem is closely related to the power set.

Similarly, the set of non-empty subsets of S might be denoted by P≥1(S) or P+(S).

A set can be regarded as an algebra having no nontrivial operations or defining equations.

The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set.

So in that regard, subalgebras behave analogously to subsets.

However, there are two important properties of subsets that do not carry over to subalgebras in general.

First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice.

The first property is more common; the case of having both is relatively rare.

Given two multigraphs G and H, a homomorphism h : G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges.

The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set.

Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices.

We can therefore organize the subgraphs of G as the multigraph ΩG, called the power object of G. What is special about a multigraph as an algebra is that its operations are unary.

A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s, t : E → V giving the source (start) and target (end) vertices of each edge.

An algebra all of whose operations are unary is called a presheaf.

Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier.

Although the term "power object" is sometimes used synonymously with exponential object YX, in topos theory Y is required to be Ω.

The contravariant power set functor is different from the covariant version in that it sends f to the preimage morphism, so that if

, which takes morphisms from b to c and takes them to morphisms from a to c, through b via h. [4] In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.