Complement (set theory)

In set theory, the complement of a set A, often denoted by

[2] When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A.

The relative complement of A with respect to a set B, also termed the set difference of B and A, written

is the set of elements in B that are not in A.

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined).

In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U:[3]

The absolute complement of A is usually denoted by

The following identities capture important properties of absolute complements: De Morgan's laws:[5] Complement laws:[5] Involution or double complement law: Relationships between relative and absolute complements: Relationship with a set difference: The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, A∁} is a partition of U.

If A and B are sets, then the relative complement of A in B,[5] also termed the set difference of B and A,[6] is the set of elements in B but not in A.

The relative complement of A in B is denoted

but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements

The following identities capture notable properties of relative complements: A binary relation

is defined as a subset of a product of sets

is the set complement of

The complement of relation

is often viewed as a logical matrix with rows representing the elements of

Producing the complementary relation to

then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

In the LaTeX typesetting language, the command \setminus[7] is usually used for rendering a set difference symbol, which is similar to a backslash symbol.

When rendered, the \setminus command looks identical to \backslash, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}.

A variant \smallsetminus is available in the amssymb package, but this symbol is not included separately in Unicode.

(It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT.)