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| [[Venn diagram]] of <math>~A \triangle B</math><br>
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| The symmetric difference is <br> | |
| the [[Union (set theory)|union]] [[Complement (set theory)#Relative complement|without]] the [[Intersection (set theory)|intersection]]:<br>
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| [[File:Venn0111.svg|40px]] <math>~\setminus~</math> [[File:Venn0001.svg|40px]] <math>~=~</math> [[File:Venn0110.svg|40px]]
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| ]]
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| In [[mathematics]], the '''symmetric difference''' of two [[Set (mathematics)|sets]] is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by
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| :<math> A\,\triangle\,B\,</math>
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| or
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| :<math>A \ominus B.</math>
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| For example, the symmetric difference of the sets <math>\{1,2,3\}</math> and <math>\{3,4\}</math> is <math>\{1,2,4\}</math>. The symmetric difference of the set of all students and the set of all females consists of all male students together with all female non-students.
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| The [[power set]] of any set becomes an [[abelian group]] under the operation of symmetric difference, with the [[empty set]] as the [[neutral element]] of the group and every element in this group being its own [[inverse element|inverse]]. The power set of any set becomes a [[Boolean ring]] with symmetric difference as the addition of the ring and [[intersection (set theory)|intersection]] as the multiplication of the ring.
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| == Properties ==
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| [[File:Venn 0110 1001.svg|thumb|
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| Venn diagram of <math>~A \triangle B \triangle C</math><br><br>
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| [[File:Venn 0110 0110.svg|40px]] <math>~\triangle~</math> [[File:Venn 0000 1111.svg|40px]] <math>~=~</math> [[File:Venn 0110 1001.svg|40px]]
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| ]]
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| The symmetric difference is equivalent to the [[union (set theory)|union]] of both [[complement (set theory)|relative complement]]s, that is:
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| :<math>A\,\triangle\,B = (A \smallsetminus B) \cup (B \smallsetminus A),\,</math>
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| and it can also be expressed as the union of the two sets, minus their [[intersection (set theory)|intersection]]:
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| :<math>A\,\triangle\,B = (A \cup B) \smallsetminus (A \cap B),</math>
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| or with the [[Exclusive or|XOR]] operation:
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| :<math>A\,\triangle\,B = \{x : (x \in A) \oplus (x \in B)\}.</math>
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| In particular, <math>A\triangle B\subseteq A\cup B</math>.
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| The symmetric difference is [[commutativity|commutative]] and [[associativity|associative]]:
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| :<math>A\,\triangle\,B = B\,\triangle\,A,\,</math>
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| :<math>(A\,\triangle\,B)\,\triangle\,C = A\,\triangle\,(B\,\triangle\,C).\,</math>
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| Thus, the repeated symmetric difference is an operation on a [[multiset]] of sets giving the set of elements which are in an odd number of sets.
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| The symmetric difference of two repeated symmetric differences is the repeated symmetric difference of the [[Multiset#Operations|join]] of the two multisets, where for each double set both can be removed. In particular:
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| :<math>(A\,\triangle\,B)\,\triangle\,(B\,\triangle\,C) = A\,\triangle\,C.\,</math>
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| This implies a sort of [[triangle inequality]]: the symmetric difference of ''A'' and ''C'' is contained in the union of the symmetric difference of ''A'' and ''B'' and that of ''B'' and ''C''. (But note that for the [[diameter]] of the symmetric difference the triangle inequality does not hold.)
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| The [[empty set]] is [[identity element|neutral]], and every set is its own inverse:
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| :<math>A\,\triangle\,\varnothing = A,\,</math>
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| :<math>A\,\triangle\,A = \varnothing.\,</math>
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| Taken together, we see that the [[power set]] of any set ''X'' becomes an [[abelian group]] if we use the symmetric difference as operation. Because every element in this group is its own inverse, this is in fact a [[vector space]] over the [[finite field|field with 2 elements]] '''Z'''<sub>2</sub>. If ''X'' is finite, then the [[singleton (mathematics)|singleton]]s form a [[basis (linear algebra)|basis]] of this vector space, and its [[Hamel dimension|dimension]] is therefore equal to the number of elements of ''X''. This construction is used in [[graph theory]], to define the [[cycle space]] of a graph.
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| Intersection [[distributivity|distributes]] over symmetric difference:
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| :<math>A \cap (B\,\triangle\,C) = (A \cap B)\,\triangle\,(A \cap C),</math>
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| and this shows that the power set of ''X'' becomes a [[ring (mathematics)|ring]] with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a [[Boolean ring]]. | |
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| Further properties of the symmetric difference:
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| * <math>A\triangle B=A^c\triangle B^c</math>, where <math>A^c</math>,<math>B^c</math> is <math>A</math>'s complement,<math>B</math>'s complement, respectively, relative to any (fixed) set that contains both.
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| * <math>\left(\bigcup_{\alpha\in\mathcal{I}}A_\alpha\right)\triangle\left(\bigcup_{\alpha\in\mathcal{I}}B_\alpha\right)\subseteq\bigcup_{\alpha\in\mathcal{I}}\left(A_\alpha\triangle B_\alpha\right)</math>, where <math>\mathcal{I}</math> is an arbitrary non-empty index set.
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| The symmetric difference can be defined in any [[Boolean algebra (structure)|Boolean algebra]], by writing
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| :<math> x\,\triangle\,y = (x \lor y) \land \lnot(x \land y) = (x \land \lnot y) \lor (y \land \lnot x) = x \oplus y.</math> | |
| This operation has the same properties as the symmetric difference of sets.
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| ==''n''-ary symmetric difference==
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| As above, the symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection:
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| :<math>\triangle M = \left\{ a \in \bigcup M: |\{A\in M:a \in A\}| \mbox{ is odd}\right\}</math>.
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| Evidently, this is well-defined only when each element of the union <math>\bigcup M</math> is contributed by a finite number of elements of <math>M</math>.
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| Suppose <math>M=\{M_{1},M_{2}, \ldots , M_{n}\}</math> is a [[multiset]] and <math>n \ge 2</math>. Then there is a formula for <math>|\triangle M|</math>, the number of elements in <math>\triangle M</math>, given solely in terms of intersections of elements of <math>M</math>:
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| :<math>|\triangle M| = \sum_{l=1}^{n} (-2)^{l-1} \sum_{i_{1} \ne i_{2} \ne \ldots \ne i_{l}} |M_{i_{1}} \cap M_{i_{2}} \cap \ldots \cap M_{i_{l}}|</math>, | |
| where <math>i_{1} \ne i_{2} \ne \ldots \ne i_{l}</math> is meant to indicate that <math>\{i_{1}, i_{2}, \ldots, i_{l}\}</math> is a subset of distinct elements of <math>\{1,2,\ldots,n\}</math>, of which there are <math>\binom{n}{l}</math>.
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| ==Symmetric difference on measure spaces==
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| As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. Formally, if μ is a [[Sigma-finite|σ-finite]] [[measure space|measure]] defined on a [[sigma-algebra|σ-algebra]] Σ, the function
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| :<math>d(X,Y) = \mu(X\,\triangle\,Y)</math>
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| is a [[Pseudometric space|pseudometric]] on Σ. ''d'' becomes a [[metric space|metric]] if Σ is considered modulo the [[equivalence relation]] ''X'' ~ ''Y'' if and only if <math>\mu(X\,\triangle\,Y) = 0</math>. The resulting metric space is [[Separable space|separable]] if and only if [[L^2|L<sup>2</sup>(μ)]] is separable.
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| Let <math>S=\left(\Omega,\mathcal{A},\mu\right)</math> be some measure space and let <math>F,G\in\mathcal{A}</math> and <math>\mathcal{D},\mathcal{E}\subseteq\mathcal{A}</math>.
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| Symmetric difference is measurable: <math>F\triangle G\in\mathcal{A}</math>.
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| We write <math>F=G\left[\mathcal{A},\mu\right]</math> iff <math>\mu\left(F\triangle G\right)=0</math>. The relation "<math>=\left[\mathcal{A},\mu\right]</math>" is an equivalence relation on the <math>\mathcal{A}</math>-measurable sets.
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| We write <math>\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A},\mu\right]</math> iff to each <math>D\in\mathcal{D}</math> there's some <math>E\in\mathcal{E}</math> such that <math>D=E\left[\mathcal{A},\mu\right]</math>. The relation "<math>\subseteq\left[\mathcal{A},\mu\right]</math>" is a partial order on the family of subsets of <math>\mathcal{A}</math>.
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| We write <math>\mathcal{D}=\mathcal{E}\left[\mathcal{A},\mu\right]</math> iff <math>\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A},\mu\right]</math> and <math>\mathcal{E}\subseteq\mathcal{D}\left[\mathcal{A},\mu\right]</math>. The relation "<math>=\left[\mathcal{A},\mu\right]</math>" is an equivalence relationship between the subsets of <math>\mathcal{A}</math>.
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| The "symmetric closure" of <math>\mathcal{D}</math> is the collection of all <math>\mathcal{A}</math>-measurable sets that are <math>=\left[\mathcal{A},\mu\right]</math> to some <math>D\in\mathcal{D}</math>. The symmetric closure of <math>\mathcal{D}</math> contains <math>\mathcal{D}</math>. If <math>\mathcal{D}</math> is a sub-<math>\sigma</math>-algebra of <math>\mathcal{A}</math>, so is the symmetric closure of <math>\mathcal{D}</math>.
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| <math>F=G\left[\mathcal{A},\mu\right]</math> iff <math>\left|\mathbf{1}_F-\mathbf{1}_G\right|=0</math> <math>\left[\mathcal{A},\mu\right]</math>-a.e.
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| ==See also==
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| {{col-begin}}
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| {{col-break}}
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| * [[Algebra of sets]]
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| * [[Boolean function]]
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| * [[Difference (set theory)]]
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| * [[Exclusive or]]
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| * [[Fuzzy set]]
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| {{col-break}}
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| * [[Logical graph]]
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| * [[Set theory]]
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| * [[Symmetry]]
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| {{col-end}}
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| ==References==
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| * {{cite book | last=Halmos | first=Paul R. | authorlink=Paul Halmos | title=[[Naive Set Theory (book)|Naive set theory]] | series=The University Series in Undergraduate Mathematics | publisher=van Nostrand Company | year=1960 | zbl=0087.04403 }}
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| * {{planetmath reference|id=916|title=Symmetric difference}}
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| * {{MathWorld |title=Symmetric Difference |urlname=SymmetricDifference}}
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| * [http://www.encyclopediaofmath.org/index.php/Symmetric_difference_of_sets ''Symmetric difference of sets'']. In [[Encyclopaedia of Mathematics]]
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| {{Set theory}}
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| [[Category:Algebra]]
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| [[Category:Basic concepts in set theory]]
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| [[Category:Binary operations]]
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