In set theory, the complement of a set , often denoted by (or ), are the elements not in .
When all sets under consideration are considered to be subsets of a given set , the absolute complement of is the set of elements in , but not in .
The relative complement of with respect to a set , also termed the set difference of and , written , is the set of elements in but not in .

** Absolute complement **

** Definition **

If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in :
: $A^c\; =\; U-A$.
Or formally:
: $A^c\; =\; \backslash .$
The absolute complement of is usually denoted by . Other notations include and .

** Examples **

* Assume that the universe is the set of integers. If is the set of odd numbers, then the complement of is the set of even numbers. If is the set of multiples of 3, then the complement of is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
* Assume that the universe is the standard 52-card deck. If the set is the suit of spades, then the complement of is the union of the suits of clubs, diamonds, and hearts. If the set is the union of the suits of clubs and diamonds, then the complement of is the union of the suits of hearts and spades.

** Properties **

Let and be two sets in a universe . The following identities capture important properties of absolute complements:
De Morgan's laws:
* $\backslash left(A\; \backslash cup\; B\; \backslash right)^c=A^c\; \backslash cap\; B^c.$
* $\backslash left(A\; \backslash cap\; B\; \backslash right)^c=A^c\backslash cup\; B^c.$
Complement laws:
* $A\; \backslash cup\; A^c\; =\; U\; .$
* $A\; \backslash cap\; A^c\; =\backslash varnothing\; .$
* $\backslash varnothing^c\; =U.$
* $U^c\; =\backslash varnothing.$
* $\backslash textA\backslash subseteq\; B\backslash textB^c\backslash subseteq\; A^c.$
*: (this follows from the equivalence of a conditional with its contrapositive).
Involution or double complement law:
* $\backslash left(A^c\backslash right)^c\; =\; A.$
Relationships between relative and absolute complements:
* $A\; \backslash setminus\; B\; =\; A\; \backslash cap\; B^c.$
* $(A\; \backslash setminus\; B)^c\; =\; A^c\; \backslash cup\; B\; =\; A^c\; \backslash cup\; (B\; \backslash cap\; A).$
Relationship with a set difference:
* $A^c\; \backslash setminus\; B^c\; =\; B\; \backslash setminus\; A.$
The first two complement laws above show that if is a non-empty, proper subset of , then is a partition of .

** Relative complement **

** Definition **

If and are sets, then the relative complement of in ,. also termed the set difference of and ,. is the set of elements in but not in .
The relative complement of in is denoted according to the ISO 31-11 standard. It is sometimes written , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements , where is taken from and from .
Formally:
: $B\; \backslash setminus\; A\; =\; \backslash .$

** Examples **

* $\backslash \; \backslash setminus\; \backslash \; =\; \backslash $.
* $\backslash \; \backslash setminus\; \backslash \; =\; \backslash $.
* If $\backslash mathbb$ is the set of real numbers and $\backslash mathbb$ is the set of rational numbers, then $\backslash mathbb\backslash setminus\backslash mathbb$ is the set of irrational numbers.

** Properties **

Let , , and be three sets. The following identities capture notable properties of relative complements:
:* $C\; \backslash setminus\; (A\; \backslash cap\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cup\; (C\; \backslash setminus\; B)$.
:* $C\; \backslash setminus\; (A\; \backslash cup\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cap\; (C\; \backslash setminus\; B)$.
:* $C\; \backslash setminus\; (B\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)\; \backslash cup\; (C\; \backslash setminus\; B)$,
:*:with the important special case $C\; \backslash setminus\; (C\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)$ demonstrating that intersection can be expressed using only the relative complement operation.
:* $(B\; \backslash setminus\; A)\; \backslash cap\; C\; =\; (B\; \backslash cap\; C)\; \backslash setminus\; A\; =\; B\; \backslash cap\; (C\; \backslash setminus\; A)$.
:* $(B\; \backslash setminus\; A)\; \backslash cup\; C\; =\; (B\; \backslash cup\; C)\; \backslash setminus\; (A\; \backslash setminus\; C)$.
:* $A\; \backslash setminus\; A\; =\; \backslash empty$.
:* $\backslash empty\; \backslash setminus\; A\; =\; \backslash empty$.
:* $A\; \backslash setminus\; \backslash empty\; =\; A$.
:* $A\; \backslash setminus\; U\; =\; \backslash empty$.

** Complementary relation **

A binary relation ''R'' is defined as a subset of a product of sets ''X'' × ''Y''. The complementary relation $\backslash bar$ is the set complement of ''R'' in ''X'' × ''Y''. The complement of relation ''R'' can be written
:$\backslash bar\; \backslash \; =\; \backslash \; (X\; \backslash times\; Y)\; \backslash setminus\; R\; .$
Here, ''R'' is often viewed as a logical matrix with rows representing the elements of ''X'', and columns elements of ''Y''. The truth of ''aRb'' corresponds to 1 in row ''a'', column ''b''. Producing the complementary relation to ''R'' 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.

** LaTeX notation **

In the LaTeX typesetting language, the command

The Comprehensive LaTeX Symbol List is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the

** In programming languages **

Some programming languages have sets among their builtin data structures. Such a data structure behaves as a finite set, that is, it consists of a finite number of data that are not specifically ordered, and may thus be considered as the elements of a set. In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. These programming languages have operators or functions for computing the complement and the set differences.
These operators may generally be applied also to data structures that are not really mathematical sets, such as ordered lists or arrays. It follows that some programming languages may have a function called

** See also **

* Algebra of sets
* Intersection (set theory)
* List of set identities and relations
* Naive set theory
* Symmetric difference
* Union (set theory)

** Notes **

** References **

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** External links **

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{{DEFAULTSORT:Complement (set theory)
Category:Operations on sets

`\setminus`

The Comprehensive LaTeX Symbol List 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`

. A variant `\smallsetminus`

is available in the amssymb package.
`set_difference`

, even if they do not have any data structure for sets.