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| In [[set theory]], a '''complement''' of a set ''A'' refers to things not in (that is, things outside of) ''A''. The '''relative complement''' of ''A'' with respect to a set ''B'', is the set of elements in ''B'' but not in ''A''. When all sets under consideration are considered to be subsets of a given set ''U'', the '''absolute complement''' of ''A'' is the set of all elements in ''U'' but not in ''A''.
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| == Relative complement ==
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| <!-- Many links redirect to this section:
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| [[difference (set theory)]], [[difference of two sets]], [[relative complement]], [[set-theoretic difference]], [[set difference]], [[set minus]], [[set subtraction]], [[set theoretic difference]], [[setminus]] -->
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| If ''A'' and ''B'' are [[set (mathematics)|sets]], then the '''relative complement''' of ''A'' in ''B'',<ref name="H17">Halmos (1960) p.17</ref> also termed the '''set-theoretic difference''' of ''B'' and ''A'',<ref>Devlin (1979) p.6</ref> is the set of elements in ''B'', but not in ''A''.
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| [[File:Venn0010.svg|250px|thumb|The '''relative complement''' of ''A'' (left circle) in ''B'' (right circle):
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| <math>A^c \cap B~~~~=~~~~B \smallsetminus A</math>]]
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| The relative complement of ''A'' in ''B'' is denoted {{nowrap|''B'' ∖ ''A''}} according to the [[ISO 31-11#Sets|ISO 31-11 standard]] (sometimes written {{nowrap|''B'' – ''A''}}, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all {{nowrap|''b'' – ''a''}}, where ''b'' is taken from ''B'' and ''a'' from ''A'').
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| Formally
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| : <math>B \smallsetminus A = \{ x\in B \, | \, x \notin A \}. </math>
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| Examples:
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| :* {1,2,3} ∖ {2,3,4} = {1}
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| :* {2,3,4} ∖ {1,2,3} = {4}
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| :* If <math>\mathbb{R}</math> is the set of [[real number]]s and <math>\mathbb{Q}</math> is the set of [[rational number]]s, then <math> \mathbb{R}\smallsetminus\mathbb{Q} = \mathbb{J} </math> is the set of [[irrational number]]s.
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| The following lists some notable properties of relative complements in relation to the set-theoretic [[operation (mathematics)|operations]] of [[union (set theory)|union]] and [[intersection (set theory)|intersection]].
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| If ''A'', ''B'', and ''C'' are sets, then the following [[identity (mathematics)|identities]] hold:
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| :* ''C'' ∖ (''A'' ∩ ''B'') = (''C'' ∖ ''A'')∪(''C'' ∖ ''B'')
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| :* ''C'' ∖ (''A'' ∪ ''B'') = (''C'' ∖ ''A'')∩(''C'' ∖ ''B'')
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| :* ''C'' ∖ (''B'' ∖ ''A'') = (''C'' ∩ ''A'')∪(''C'' ∖ ''B'')
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| [ Alternately written: ''A'' ∖ (''B'' ∖ ''C'') = (''A'' ∖ ''B'')∪(''A'' ∩ ''C'') ]
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| :* (''B'' ∖ ''A'') ∩ ''C'' = (''B'' ∩ ''C'') ∖ ''A'' = ''B''∩(''C'' ∖ ''A'')
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| :* (''B'' ∖ ''A'') ∪ ''C'' = (''B'' ∪ ''C'') ∖ (''A'' ∖ ''C'')
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| :* ''A'' ∖ ''A'' = Ø
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| :* Ø ∖ ''A'' = Ø
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| :* ''A'' ∖ Ø = ''A''
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| ==Absolute complement==<!-- This section is linked from [[Bayes' theorem]] and [[absolute set complement]] -->
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| [[File:Venn1010.svg|250px|thumb|The '''absolute complement''' of A in U:
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| <math>A^c=U \smallsetminus A</math>]]
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| If a [[universe (mathematics)|universe]] '''U''' is defined, then the relative complement of ''A'' in '''U''' is called the '''absolute complement''' (or simply '''complement''') of ''A'', and is denoted by ''A''<sup>c</sup> or sometimes ''A''′, also the same set often{{citation needed|date=August 2012}} is denoted by <MATH>\complement_U A</MATH> or <MATH>\complement A</MATH> if '''U''' is fixed, that is:
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| : ''A''<sup>c</sup> = '''U''' ∖ ''A''.
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| For example, if the universe is the set of [[integer]]s, then the complement of the set of odd numbers is the set of even numbers.
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| The following lists some important properties of absolute complements in relation to the set-theoretic [[operation (mathematics)|operations]] of union and [[intersection (set theory)|intersection]].
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| If ''A'' and ''B'' are subsets of a [[universe (mathematics)|universe]] '''U''', then the following identities hold:
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| : [[De Morgan's laws]]:<ref name="H17" />
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| ::* <math>\left(A \cup B \right)^{c}=A^{c} \cap B^{c} .</math>
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| ::* <math>\left(A \cap B \right)^{c}=A^{c} \cup B^{c} .</math>
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| : Complement laws:<ref name="H17" />
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| ::* <math>A \cup A^{c} =U .</math>
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| ::* <math>A \cap A^{c} =\empty .</math>
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| ::* <math>\empty ^{c} =U.</math>
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| ::* <math> U^{c} =\empty.</math>
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| ::* <math>\text{If }A\subset B\text{, then }B^{c}\subset A^{c}.</math>
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| ::*: (this follows from the equivalence of a conditional with its [[contrapositive]])
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| : [[involution (mathematics)|Involution]] or double complement law:
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| ::* <math>\left(A^{c}\right)^{c}=A.</math>
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| : Relationships between relative and absolute complements:
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| ::* ''A'' ∖ ''B'' = ''A'' ∩ ''B''<sup>c</sup>
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| ::* (''A'' ∖ ''B'')<sup>c</sup> = ''A''<sup>c</sup> ∪ ''B''
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| The first two complement laws above shows that if ''A'' is a non-empty, [[proper subset]] of '''U''', then {''A'', ''A''<sup>c</sup>} is a [[partition of a set|partition]] of '''U'''.
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| == Notation ==
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| In the [[LaTeX]] typesetting language, the command <code>\setminus</code> is usually used for rendering a set difference symbol, which is similar to a [[backslash]] symbol. When rendered the <code>\setminus</code> command looks identical to <code>\backslash</code> except that it has a little more space in front and behind the slash, akin to the LaTeX sequence <code>\mathbin{\backslash}</code>. A variant <code>\smallsetminus</code> is available in the amssymb package.
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| == Complements in various programming languages ==
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| Some programming languages allow for manipulation of [[set (computer science)|sets as data structures]], using these operators or functions to construct the difference of sets <code>a</code> and <code>b</code>:
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| <!-- please keep in alphabetical order of programming language name -->
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| ; [[.NET Framework]]
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| : <code>a.Except(b);</code>
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| ; [[C++]]
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| : <code>set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());</code>
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| ; [[Clojure]]
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| : <code>(clojure.set/difference a b)</code><ref>[http://richhickey.github.com/clojure/clojure.set-api.html#clojure.set/difference] clojure.set API reference</ref>
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| ; [[Common Lisp]]
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| : <code>set-difference, nset-difference</code><ref name="CLHS_set-difference">[http://www.lispworks.com/documentation/HyperSpec/Body/f_set_di.htm Common Lisp HyperSpec, Function set-difference, nset-difference]. Accessed on September 8, 2009.</ref>
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| ; [[Haskell (programming language)|Haskell]]
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| : <code>a \\ b</code> <ref name="Data.Set">[http://haskell.org/ghc/docs/latest/html/libraries/containers/Data-Set.html Data.Set (Haskell)]</ref>
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| ; [[Java (programming language)|Java]]
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| : <code>diff = a.clone();
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| : diff.removeAll(b);</code><ref name="J2SE_Set">[http://java.sun.com/j2se/1.5.0/docs/api/java/util/Set.html Set (Java 2 Platform SE 5.0)]. ''JavaTM 2 Platform Standard Edition 5.0 API Specification'', updated in 2004. Accessed on February 13, 2008.</ref>
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| ; [[Mathematica]]
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| : <code>Complement</code><ref name="Mathematica_set">[http://reference.wolfram.com/mathematica/ref/Complement.html Complement]. ''Mathematica Documentation Center'' for version 6.0, updated in 2008. Accessed on March 7, 2008.</ref>
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| ; [[MATLAB]]
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| : <code>setdiff</code><ref name="MATLAB_set">[http://www.mathworks.com/access/helpdesk/help/techdoc/ref/setdiff.html Setdiff]. ''MATLAB Function Reference'' for version 7.6, updated in 2008. Accessed on May 19, 2008.</ref>
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| ; [[OCaml]]
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| : <code>Set.S.diff</code><ref name="OCaml_Set_S_diff">[http://caml.inria.fr/pub/docs/manual-ocaml/libref/Set.S.html Set.S (OCaml)].</ref>
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| ; [[GNU Octave|Octave]]
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| : <code>setdiff</code><ref>[http://www.gnu.org/software/octave/doc/interpreter/index.html]. ''GNU Octave Reference Manual''</ref>
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| ; [[Pascal (programming language)|Pascal]]
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| : <code>SetDifference := a - b;</code>
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| ; [[Perl 5]]
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| : <code>#for perl version >= 5.10</code>
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| : <code>@a = grep {not $_ ~~ @b} @a;</code>
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| ; [[Perl 6]]
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| : <code>$A ∖ $B</code>
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| : <code>$A (-) $B # texas version</code>
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| ; [[PHP]]
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| : <code>array_diff($a, $b);</code><ref>[http://php.net/manual/en/function.array-diff.php PHP: array_diff], PHP Manual</ref>
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| ; [[Prolog]]
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| : <code>a(X),\+ b(X).</code>
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| ; [[Python (programming language)|Python]]
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| : <code>diff = a.difference(b)</code><ref name="Python_set">[http://docs.python.org/2/library/stdtypes.html?highlight=difference#set.difference]. ''Python v2.7.3 documentation''. Accessed on January 17, 2013.</ref>
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| : <code>diff = a - b</code><ref name="Python_set" />
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| ; [[R (programming language)|R]]
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| : <code>setdiff</code><ref>[http://cran.r-project.org/doc/manuals/fullrefman.pdf R Reference manual p. 410].</ref>
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| ; [[Ruby (programming language)|Ruby]]
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| : <code>diff = a - b</code><ref>[http://www.ruby-doc.org/core/classes/Array.html Class: Array] Ruby Documentation</ref>
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| ; [[Scala (programming language)|Scala]]
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| : <code>diff = a—b</code><ref name="Scala_set">[http://www.scala-lang.org/api/current scala.collection.Set]. ''Scala Standard Library'' release 2.8.1, Accessed on December 09, 2010.</ref>
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| ; [[Smalltalk (Pharo)]]
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| : <code>a difference: b</code>
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| ; [[SQL]]
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| : <code> SELECT * FROM A
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| MINUS
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| SELECT * FROM B
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| </code>
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| ; [[Unix shell]]
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| : <code>comm -23 a b</code><ref name="Unix_comm">[http://plan9.bell-labs.com/7thEdMan/ comm(1)], Unix Seventh Edition Manual, 1979.</ref>
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| : <code>grep -vf b a</code> # less efficient, but works with small unsorted sets
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| == See also ==
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| * [[Algebra of sets]]
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| * [[Naive set theory]]
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| * [[Symmetric difference]]
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| == References ==
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| <references/>
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| * {{cite book | last=Halmos | first=Paul R. | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=van Nostrand Company | year=1960 | zbl=0087.04403 }}
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| * {{cite book | last=Devlin | first=Keith J. | authorlink=Keith Devlin | title=Fundamentals of contemporary set theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | zbl=0407.04003 }}
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| == External links ==
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| * {{MathWorld |title=Complement |id=Complement }}
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| * {{MathWorld |title=Complement Set |id=ComplementSet }}
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| {{Set theory}}
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| {{DEFAULTSORT:Complement (set theory)}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Binary operations]]
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