Complement (set theory)

<h2 id="absolute-complement">Absolute complement</h2>

<h3>Definition</h3>
<p>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:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>

A
          
            c
          
        
        =
        U
        ∖
        A
        =
        {
        x
        ∈
        U
        :
        x
        ∉
        A
        }
        .
      
    
    {\displaystyle A^{c}=U\setminus A=\{x\in U:x\notin A\}.}

</p><p>The absolute complement of A is usually denoted by 
  
    
      
        
          A
          
            c
          
        
      
    
    {\displaystyle A^{c}}
  
. Other notations include 
  
    
      
        
          
            A
            ¯
          
        
        ,
        
          A
          ′
        
        ,
      
    
    {\displaystyle {\overline {A}},A',}
  
<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> 
  
    
      
        
          ∁
          
            U
          
        
        A
        ,
        
           and 
        
        ∁
        A
        .
      
    
    {\displaystyle \complement _{U}A,{\text{ and }}\complement A.}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Examples</h3>
<ul><li>Assume that the universe is the set of <a href="/facts/Integer/PUwwrawx">integers</a>. If A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of <a href="/facts/Multiple_(mathematics)/tkte6NXa">multiples</a> of 3, then the complement of B is the set of numbers <a href="/facts/Modular_arithmetic/0wtRa5dU">congruent</a> to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).</li>
<li>Assume that the universe is the <a href="/facts/Standard_52-card_deck/VKrPQBHf">standard 52-card deck</a>. If the set A is the suit of spades, then the complement of A is the <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts and spades.</li>
<li>When the universe is the <a href="/facts/Universe_(mathematics)/v2oJnu4e">universe of sets</a> described in formalized <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the absolute complement of a set is generally not itself a set, but rather a <a href="/facts/Proper_class/F5EmDd9b">proper class</a>.  For more info, see <a href="/facts/Universal_set/QLFgSfox">universal set</a>.</li></ul>
<h3>Properties</h3>
<p>Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:
</p><p><a href="/facts/De_Morgan%27s_laws/dj3F8xpt">De Morgan's laws</a>:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<ul><li>
  
    
      
        
          
            (
            
              A
              ∪
              B
            
            )
          
          
            c
          
        
        =
        
          A
          
            c
          
        
        ∩
        
          B
          
            c
          
        
        .
      
    
    {\displaystyle \left(A\cup B\right)^{c}=A^{c}\cap B^{c}.}
  
</li>
<li>
  
    
      
        
          
            (
            
              A
              ∩
              B
            
            )
          
          
            c
          
        
        =
        
          A
          
            c
          
        
        ∪
        
          B
          
            c
          
        
        .
      
    
    {\displaystyle \left(A\cap B\right)^{c}=A^{c}\cup B^{c}.}
  
</li></ul>
<p>Complement laws:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<ul><li>
  
    
      
        A
        ∪
        
          A
          
            c
          
        
        =
        U
        .
      
    
    {\displaystyle A\cup A^{c}=U.}
  
</li>
<li>
  
    
      
        A
        ∩
        
          A
          
            c
          
        
        =
        ∅
        .
      
    
    {\displaystyle A\cap A^{c}=\emptyset .}
  
</li>
<li>
  
    
      
        
          ∅
          
            c
          
        
        =
        U
        .
      
    
    {\displaystyle \emptyset ^{c}=U.}
  
</li>
<li>
  
    
      
        
          U
          
            c
          
        
        =
        ∅
        .
      
    
    {\displaystyle U^{c}=\emptyset .}
  
</li>
<li>
  
    
      
        
          If 
        
        A
        ⊆
        B
        
          , then 
        
        
          B
          
            c
          
        
        ⊆
        
          A
          
            c
          
        
        .
      
    
    {\displaystyle {\text{If }}A\subseteq B{\text{, then }}B^{c}\subseteq A^{c}.}

(this follows from the equivalence of a conditional with its <a href="/facts/Contrapositive/ENPumbyv">contrapositive</a>).</li></ul>
<p><a href="/facts/Involution_(mathematics)/kKxYrUVa">Involution</a> or double complement law:
</p>
<ul><li>
  
    
      
        
          
            (
            
              A
              
                c
              
            
            )
          
          
            c
          
        
        =
        A
        .
      
    
    {\displaystyle \left(A^{c}\right)^{c}=A.}
  
</li></ul>
<p>Relationships between relative and absolute complements:
</p>
<ul><li>
  
    
      
        A
        ∖
        B
        =
        A
        ∩
        
          B
          
            c
          
        
        .
      
    
    {\displaystyle A\setminus B=A\cap B^{c}.}
  
</li>
<li>
  
    
      
        (
        A
        ∖
        B
        
          )
          
            c
          
        
        =
        
          A
          
            c
          
        
        ∪
        B
        =
        
          A
          
            c
          
        
        ∪
        (
        B
        ∩
        A
        )
        .
      
    
    {\displaystyle (A\setminus B)^{c}=A^{c}\cup B=A^{c}\cup (B\cap A).}
  
</li></ul>
<p>Relationship with a set difference:
</p>
<ul><li>
  
    
      
        
          A
          
            c
          
        
        ∖
        
          B
          
            c
          
        
        =
        B
        ∖
        A
        .
      
    
    {\displaystyle A^{c}\setminus B^{c}=B\setminus A.}
  
</li></ul>
<p>The first two complement laws above show that if <i>A</i> is a non-empty, <a href="/facts/Proper_subset/SJGERmbA">proper subset</a> of <i>U</i>, then {<i>A</i>, <i>A</i>∁} is a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of <i>U</i>.
</p>
<h2 id="relative-complement">Relative complement</h2>
<h3>Definition</h3>
<p>If <i>A</i> and <i>B</i> are sets, then the relative complement of <i>A</i> in <i>B</i>,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> also termed the set difference of <i>B</i> and <i>A</i>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> is the set of elements in <i>B</i> but not in <i>A</i>.
</p>

<p>The relative complement of <i>A</i> in <i>B</i> is denoted 
  
    
      
        B
        ∖
        A
      
    
    {\displaystyle B\setminus A}
  
 according to the <a href="/facts/ISO_31-11/IjFTjgw4">ISO 31-11 standard</a>. It is sometimes written 
  
    
      
        B
        −
        A
        ,
      
    
    {\displaystyle B-A,}
  
 but this notation is ambiguous, as in some contexts (for example, <a href="/facts/Minkowski_addition/dEQfoJ1A">Minkowski set operations</a> in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>) it can be interpreted as the set of all elements 
  
    
      
        b
        −
        a
        ,
      
    
    {\displaystyle b-a,}
  
 where <i>b</i> is taken from <i>B</i> and <i>a</i> from <i>A</i>.
</p><p>Formally:

B
        ∖
        A
        =
        {
        x
        ∈
        B
        :
        x
        ∉
        A
        }
        .
      
    
    {\displaystyle B\setminus A=\{x\in B:x\notin A\}.}

</p>
<h3>Examples</h3>
<ul><li>
  
    
      
        {
        1
        ,
        2
        ,
        3
        }
        ∖
        {
        2
        ,
        3
        ,
        4
        }
        =
        {
        1
        }
        .
      
    
    {\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}.}
  
</li>
<li>
  
    
      
        {
        2
        ,
        3
        ,
        4
        }
        ∖
        {
        1
        ,
        2
        ,
        3
        }
        =
        {
        4
        }
        .
      
    
    {\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}.}
  
</li>
<li>If 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is the set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
 is the set of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, then 
  
    
      
        
          R
        
        ∖
        
          Q
        
      
    
    {\displaystyle \mathbb {R} \setminus \mathbb {Q} }
  
 is the set of <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>.</li></ul>
<h3>Properties</h3>
<p class="note">See also: <a href="/facts/List_of_set_identities_and_relations/yekH29va">List of set identities and relations</a> and <a href="/facts/Algebra_of_sets/ytgmwAqz">Algebra of sets</a></p>
<p>Let <i>A</i>, <i>B</i>, and <i>C</i> be three sets in a universe U. The following <a href="/facts/Identity_(mathematics)/LR46j4uR">identities</a> capture notable properties of relative complements:
</p>
<ul><li>
  
    
      
        C
        ∖
        (
        A
        ∩
        B
        )
        =
        (
        C
        ∖
        A
        )
        ∪
        (
        C
        ∖
        B
        )
        .
      
    
    {\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B).}
  
</li>
<li>
  
    
      
        C
        ∖
        (
        A
        ∪
        B
        )
        =
        (
        C
        ∖
        A
        )
        ∩
        (
        C
        ∖
        B
        )
        .
      
    
    {\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B).}
  
</li>
<li>
  
    
      
        C
        ∖
        (
        B
        ∖
        A
        )
        =
        (
        C
        ∩
        A
        )
        ∪
        (
        C
        ∖
        B
        )
        ,
      
    
    {\displaystyle C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B),}

with the important special case 
  
    
      
        C
        ∖
        (
        C
        ∖
        A
        )
        =
        (
        C
        ∩
        A
        )
      
    
    {\displaystyle C\setminus (C\setminus A)=(C\cap A)}
  
 demonstrating that intersection can be expressed using only the relative complement operation.</li>
<li>
  
    
      
        (
        B
        ∖
        A
        )
        ∩
        C
        =
        (
        B
        ∩
        C
        )
        ∖
        A
        =
        B
        ∩
        (
        C
        ∖
        A
        )
        .
      
    
    {\displaystyle (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).}
  
</li>
<li>
  
    
      
        (
        B
        ∖
        A
        )
        ∪
        C
        =
        (
        B
        ∪
        C
        )
        ∖
        (
        A
        ∖
        C
        )
        .
      
    
    {\displaystyle (B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).}
  
</li>
<li>
  
    
      
        A
        ∖
        A
        =
        ∅
        .
      
    
    {\displaystyle A\setminus A=\emptyset .}
  
</li>
<li>
  
    
      
        ∅
        ∖
        A
        =
        ∅
        .
      
    
    {\displaystyle \emptyset \setminus A=\emptyset .}
  
</li>
<li>
  
    
      
        A
        ∖
        ∅
        =
        A
        .
      
    
    {\displaystyle A\setminus \emptyset =A.}
  
</li>
<li>
  
    
      
        A
        ∖
        U
        =
        ∅
        .
      
    
    {\displaystyle A\setminus U=\emptyset .}
  
</li>
<li>If 
  
    
      
        A
        ⊂
        B
      
    
    {\displaystyle A\subset B}
  
, then 
  
    
      
        C
        ∖
        A
        ⊃
        C
        ∖
        B
      
    
    {\displaystyle C\setminus A\supset C\setminus B}
  
.</li>
<li>
  
    
      
        A
        ⊇
        B
        ∖
        C
      
    
    {\displaystyle A\supseteq B\setminus C}
  
 is equivalent to 
  
    
      
        C
        ⊇
        B
        ∖
        A
      
    
    {\displaystyle C\supseteq B\setminus A}
  
.</li></ul>
<h2 id="complementary-relation">Complementary relation</h2>
<p>A <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is defined as a subset of a <a href="/facts/Product_of_sets/BfqBWzdL">product of sets</a> 
  
    
      
        X
        ×
        Y
        .
      
    
    {\displaystyle X\times Y.}
  
 The complementary relation 
  
    
      
        
          
            
              R
              ¯
            
          
        
      
    
    {\displaystyle {\bar {R}}}
  
 is the set complement of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 in 
  
    
      
        X
        ×
        Y
        .
      
    
    {\displaystyle X\times Y.}
  
 The complement of relation 
  
    
      
        R
      
    
    {\displaystyle R}
  
 can be written

R
              ¯
            
          
        
         
        =
         
        (
        X
        ×
        Y
        )
        ∖
        R
        .
      
    
    {\displaystyle {\bar {R}}\ =\ (X\times Y)\setminus R.}

Here, 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is often viewed as a <a href="/facts/Logical_matrix/pS4BnBzj">logical matrix</a> with rows representing the elements of 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 and columns elements of 
  
    
      
        Y
        .
      
    
    {\displaystyle Y.}
  
 The truth of 
  
    
      
        a
        R
        b
      
    
    {\displaystyle aRb}
  
 corresponds to 1 in row 
  
    
      
        a
        ,
      
    
    {\displaystyle a,}
  
 column 
  
    
      
        b
        .
      
    
    {\displaystyle b.}
  
 Producing the complementary relation to 
  
    
      
        R
      
    
    {\displaystyle R}
  
 then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.
</p><p>Together with <a href="/facts/Composition_of_relations/tD8joBlo">composition of relations</a> and <a href="/facts/Converse_relation/EVGlRSz7">converse relations</a>, complementary relations and the <a href="/facts/Algebra_of_sets/ytgmwAqz">algebra of sets</a> are the elementary <a href="/facts/Operation_(mathematics)/Yplv602Q">operations</a> of the <a href="/facts/Calculus_of_relations/bnkuqNb0">calculus of relations</a>.
</p>
<h2 id="latex-notation">LaTeX notation</h2>
<p class="note">See also: <a href="/facts/List_of_mathematical_symbols_by_subject/yhYvHQ66">List of mathematical symbols by subject</a></p>
<p>In the <a href="/facts/LaTeX/c1lvjF0A">LaTeX</a> typesetting language, the command \setminus<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> is usually used for rendering a set difference symbol, which is similar to a <a href="/facts/Backslash/SoV79KPp">backslash</a> 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. The symbol 
  
    
      
        ∁
      
    
    {\displaystyle \complement }
  
 (as opposed to 
  
    
      
        C
      
    
    {\displaystyle C}
  
) is produced by \complement. (It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT.)
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Algebra_of_sets/ytgmwAqz">Algebra of sets</a> – Identities and relationships involving sets</li>
<li><a href="/facts/Intersection_(set_theory)/RPfUNJh9">Intersection (set theory)</a> – Set of elements common to all of some sets</li>
<li><a href="/facts/List_of_set_identities_and_relations/yekH29va">List of set identities and relations</a> – Equalities for combinations of sets</li>
<li><a href="/facts/Naive_set_theory/hYIC5xPf">Naive set theory</a> – Informal set theories</li>
<li><a href="/facts/Symmetric_difference/W2QST1Qr">Symmetric difference</a> – Elements in exactly one of two sets</li>
<li><a href="/facts/Union_(set_theory)/KVVXKfWl">Union (set theory)</a> – Set of elements in any of some sets</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, N.</a> (1970). <i>Théorie des ensembles</i> (in French). Paris: Hermann. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-34034-8.</li>
<li><a href="/facts/Keith_Devlin/9YsdjXjk">Devlin, Keith J.</a> (1979). <i>Fundamentals of contemporary set theory</i>. Universitext. <a href="/facts/Springer-Verlag/nAesf6nT">Springer</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90441-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0407.04003">0407.04003</a>.</li>
<li><a href="/facts/Paul_Halmos/mR11lcap">Halmos, Paul R.</a> (1960). <a href="https://archive.org/details/naivesettheory0000halm"><i>Naive set theory</i></a>. The University Series in Undergraduate Mathematics. van Nostrand Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780442030643. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0087.04403">0087.04403</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Complement.html">"Complement"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ComplementSet.html">"Complement Set"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04. <a href="http://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm" target="_blank">http://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04. <a href="https://www.mathsisfun.com/definitions/complement-set-.html" target="_blank">https://www.mathsisfun.com/definitions/complement-set-.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04. <a href="https://www.mathsisfun.com/definitions/complement-set-.html" target="_blank">https://www.mathsisfun.com/definitions/complement-set-.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Bourbaki 1970, p. E II.6. - Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Halmos 1960, p. 17. - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Halmos 1960, p. 17. - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Halmos 1960, p. 17. - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Devlin 1979, p. 6. - Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer. ISBN 0-387-90441-7. Zbl 0407.04003. <a href="https://zbmath.org/?format=complete&q=an:0407.04003" target="_blank">https://zbmath.org/?format=complete&q=an:0407.04003</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>[1] Archived 2022-03-05 at the Wayback Machine The Comprehensive LaTeX Symbol List <a href="http://ctan.unsw.edu.au/info/symbols/comprehensive/symbols-a4.pdf" target="_blank">http://ctan.unsw.edu.au/info/symbols/comprehensive/symbols-a4.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Complement (set theory) open-in-new

Complement (set theory)