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- Complement (set theory)

In set theory, the **complement** of a set, often denoted by (or),^{[1]} ^{[2]} are the elements not in .^{[3]}

When all sets under consideration are considered to be subsets of a given set, the **absolute complement** of is the set of elements in that are 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 that are not in .

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 :^{[4]}

*A*^{c}=*U*-*A*

Or formally:

*A*^{c}=*\{**x\in**U**\mid**x**\notin**A**\}.*

The absolute complement of is usually denoted by . Other notations include and ^{[5]}

- 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.

Let and be two sets in a universe . The following identities capture important properties of absolute complements:

*\left(A**\cup**B**\right)*^{c=A}^{c}*\cap**B*^{c.}

*\left(A**\cap**B**\right)*^{c=A}^{c\cup}*B*^{c.}

Complement laws:

*A**\cup**A*^{c}=*U**.*

*A**\cap**A*^{c}=*\varnothing**.*

*\varnothing*^{c}=*U.*

*U*^{c}=*\varnothing.*

If*A\subseteq**B,thenB*^{c\subseteq}*A*^{c.}

(this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

*\left(A*^{c\right)}^{c}=*A.*

Relationships between relative and absolute complements:

*A**\setminus**B*=*A**\cap**B*^{c.}

*(A**\setminus**B)*^{c}=*A*^{c}*\cup**B*=*A*^{c}*\cup**(B**\cap**A).*

Relationship with a set difference:

*A*^{c}*\setminus**B*^{c}=*B**\setminus**A.*

The first two complement laws above show that if is a non-empty, proper subset of, then is a partition of .

If and are sets, then the **relative complement** of in,^{[6]} also termed the **set difference** of and,^{[7]} 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**\setminus**A*=*\{**x\in**B**\mid**x**\notin**A**\}.*

*\{*1*,*2*,*3*\}**\setminus**\{*2*,*3*,*4*\}*=*\{*1*\}*

*\{*2*,*3*,*4*\}**\setminus**\{*1*,*2*,*3*\}*=*\{*4*\}*

- If

R

Q

R*\setminusQ*

See also: List of set identities and relations.

Let,, and be three sets. The following identities capture notable properties of relative complements:

*C**\setminus**(A**\cap**B)*=*(C**\setminus**A)**\cup**(C**\setminus**B)*

*C**\setminus**(A**\cup**B)*=*(C**\setminus**A)**\cap**(C**\setminus**B)*

*C**\setminus**(B**\setminus**A)*=*(C**\cap**A)**\cup**(C**\setminus**B)*

with the important special case

*C**\setminus**(C**\setminus**A)*=*(C**\cap**A)*

*(B**\setminus**A)**\cap**C*=*(B**\cap**C)**\setminus**A*=*B**\cap**(C**\setminus**A)*

*(B**\setminus**A)**\cup**C*=*(B**\cup**C)**\setminus**(A**\setminus**C)*

*A**\setminus**A*=*\empty*

*\empty**\setminus**A*=*\empty*

*A**\setminus**\empty*=*A*

*A**\setminus**U*=*\empty*

A binary relation *R* is defined as a subset of a product of sets *X* × *Y*. The **complementary relation**

*\bar{R}*

*\bar{R}* = *(X* x *Y)**\setminus**R**.*

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`

^{[8]} 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.

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 `set_difference`

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

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

- Web site: 2020-03-01. Compendium of Mathematical Symbols. 2020-09-04. Math Vault. en-US.
- Web site: Complement and Set Difference. 2020-09-04. web.mnstate.edu.
- Web site: Complement (set) Definition (Illustrated Mathematics Dictionary). 2020-09-04. www.mathsisfun.com.
- The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
- .
- .
- .
- http://ctan.unsw.edu.au/info/symbols/comprehensive/symbols-a4.pdf

- Book: Bourbaki , N. . Nicolas Bourbaki . Théorie des ensembles . Hermann . Paris . 1970 . 978-3-540-34034-8 . fr.
- Book: Devlin , Keith J. . Keith Devlin . Fundamentals of contemporary set theory . Universitext . . 1979 . 0-387-90441-7 . 0407.04003.
- Book: Halmos , Paul R. . Paul Halmos . Naive set theory . registration . The University Series in Undergraduate Mathematics . van Nostrand Company . 1960 . 0087.04403.