Editing Symmetric difference (section)

== Properties ==
[[File:Venn 0110 1001.svg|thumb|
Venn diagram of <math>~(A  \Delta B)  \Delta C</math>

{{nowrap|[[File:Venn 0110 0110.svg|40px]] <math>~ \Delta~</math> [[File:Venn 0000 1111.svg|40px]]}} <math>~=~</math> [[File:Venn 0110 1001.svg|40px]]
]]

The symmetric difference is equivalent to the [[union (set theory)|union]] of both [[complement (set theory)|relative complement]]s, that is:<ref name=":0">{{Cite web |last=Taylor |first=Courtney |date=March 31, 2019 |title=What Is Symmetric Difference in Math? |url=https://www.thoughtco.com/what-is-the-symmetric-difference-3126594 |access-date=2020-09-05 |website=ThoughtCo |language=en}}</ref>

:<math>A\, \Delta\,B = \left(A \setminus B\right) \cup \left(B \setminus A\right),</math>

The symmetric difference can also be expressed using the [[Exclusive or|XOR]] operation ⊕ on the [[Predicate (mathematical logic)|predicates]] describing the two sets in [[set-builder notation]]:

:<math>A\mathbin{ \Delta}B = \{x : (x \in A) \oplus (x \in B)\}.</math>

The same fact can be stated as the [[indicator function]] (denoted here by <math>\chi</math>) of the symmetric difference, being the XOR (or addition [[Modular arithmetic|mod 2]]) of the indicator functions of its two arguments: <math>\chi_{(A\, \Delta\,B)} = \chi_A \oplus \chi_B</math> or using the [[Iverson bracket]] notation <math>[x \in A\, \Delta\,B] = [x \in A] \oplus [x \in B]</math>.

The symmetric difference can also be expressed as the union of the two sets, minus their [[intersection (set theory)|intersection]]:

:<math>A\, \Delta\,B = (A \cup B) \setminus (A \cap B),</math><ref name=":0" />

In particular, <math>A \mathbin{ \Delta} B\subseteq A\cup B</math>; the equality in this non-strict [[Subset|inclusion]] occurs [[if and only if]] <math>A</math> and <math>B</math> are [[disjoint sets]]. Furthermore, denoting <math>D = A \mathbin{ \Delta} B</math> and <math>I = A \cap B</math>, then <math>D</math> and <math>I</math> are always disjoint, so <math>D</math> and <math>I</math> [[Partition of a set|partition]] <math>A \cup B</math>. Consequently, assuming intersection and symmetric difference as primitive operations, the union of two sets can be well ''defined'' in terms of symmetric difference by the right-hand side of the equality

:<math>A\,\cup\,B = (A\, \Delta\,B)\, \Delta\,(A \cap B)</math>.

The symmetric difference is [[commutativity|commutative]] and [[associativity|associative]]:

:<math>\begin{align}
                  A\, \Delta\,B &= B\, \Delta\,A, \\
  (A\, \Delta\,B)\, \Delta\,C &= A\, \Delta\,(B\, \Delta\,C).
\end{align}</math>

The [[empty set]] is [[identity element|neutral]], and every set is its own inverse:
:<math>\begin{align}
  A\, \Delta\,\varnothing &= A, \\
            A\, \Delta\,A &= \varnothing.
\end{align}</math>

Thus, the [[power set]] of any set ''X'' becomes an [[abelian group]] under the symmetric difference operation. (More generally, any [[field of sets]] forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has [[Order (group theory)|order]] 2) is sometimes called a [[Boolean group]];<ref name="GivantHalmos2009">{{cite book |last1=Givant |first1=Steven |last2=Halmos |first2=Paul |author-link2=Paul Halmos |year=2009 |title=Introduction to Boolean Algebras |publisher=Springer Science & Business Media |isbn=978-0-387-40293-2 |page=6}}</ref><ref name="Humberstone2011">{{cite book |last=Humberstone |first=Lloyd |year=2011 |title=The Connectives |url=https://archive.org/details/connectives00humb |url-access=limited |publisher=MIT Press |isbn=978-0-262-01654-4 |page=[https://archive.org/details/connectives00humb/page/n800 782]}}</ref> the symmetric difference provides a prototypical example of such groups. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set.<ref name="Rotman2010">{{cite book |last=Rotman |first=Joseph J. |year=2010 |title=Advanced Modern Algebra |publisher=American Mathematical Soc. |isbn=978-0-8218-4741-1 |page=19}}</ref> In the case where ''X'' has only two elements, the group thus obtained is the [[Klein four-group]].

Equivalently, a Boolean group is an [[elementary abelian group|elementary abelian 2-group]]. Consequently, the group induced by the symmetric difference is in fact a [[vector space]] over the [[GF(2)|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 [[dimension (vector space)|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.

From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the [[Multiset#Operations|join]] of the two multisets, where for each double set both can be removed. In particular:

:<math>(A\, \Delta\,B)\, \Delta\,(B\, \Delta\,C) = A\, \Delta\,C.</math>

This implies triangle inequality:<ref>{{cite book |last1=Rudin |first1=Walter |date=January 1, 1976 |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |edition=3rd |publisher=McGraw-Hill Education |isbn=978-0070542358 |page=[https://archive.org/details/principlesofmath00rudi/page/306 306]}}</ref> 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''.

Intersection [[distributivity|distributes]] over symmetric difference:
:<math>A \cap (B\, \Delta\,C) = (A \cap B)\, \Delta\,(A \cap C),</math>

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]].

Further properties of the symmetric difference include:

* <math>A \mathbin{ \Delta} B = \emptyset</math> if and only if <math>A = B</math>.
* <math>A \mathbin{ \Delta} B = A^c \mathbin{ \Delta} 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.
* <math>\left(\bigcup_{\alpha\in\mathcal{I}}A_\alpha\right) \Delta\left(\bigcup_{\alpha\in\mathcal{I}}B_\alpha\right)\subseteq\bigcup_{\alpha\in\mathcal{I}}\left(A_\alpha \mathbin{ \Delta} B_\alpha\right)</math>, where <math>\mathcal{I}</math> is an arbitrary non-empty index set.
* If <math>f : S \rightarrow T</math> is any function and <math>A, B \subseteq T</math> are any sets in <math>f</math>'s codomain, then <math>f^{-1}\left(A \mathbin{ \Delta} B\right) = f^{-1}\left(A\right) \mathbin{ \Delta} f^{-1}\left(B\right).</math>

The symmetric difference can be defined in any [[Boolean algebra (structure)|Boolean algebra]], by writing
:<math> x\, \Delta\,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.