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Difference between revisions of "Symmetric difference of sets"

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An operation on sets. Given two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916402.png" />, their symmetric difference, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916403.png" />, is given by
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{{TEX|done}}{{MSC|03E}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916404.png" /></td> </tr></table>
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An operation on sets. Given two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by
 
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$$
where the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916408.png" /> denote the operations of union, intersection, difference, and complementation of sets, respectively.
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A \Delta B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B)
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$$
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where the symbols $\cup$, $\cap$, $\setminus$, ${}'$ denote the operations of [[union of sets|union]], [[intersection of sets|intersection]], [[Difference of two sets|difference]], and [[complementation]] of sets, respectively.
  
  
  
 
====Comments====
 
====Comments====
The symmetric difference operation is associative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916409.png" />, and intersection is distributive over it, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164010.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164012.png" /> define a ring structure on the power set of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164013.png" /> (the set of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164014.png" />), in contrast to union and intersection. This ring is the same as the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164015.png" />-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164016.png" /> (with [[pointwise multiplication]] and addition). Cf. also [[Boolean algebra]] and [[Boolean ring]] for the symmetric difference operation in an arbitrary Boolean algebra.
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The symmetric difference operation is associative, i.e. $A \Delta (B \Delta C) = (A \Delta B) \Delta C$, and intersection is distributive over it, i.e. $A \cap (B \Delta C) = (A \cap B) \Delta (A \cap C)$. Thus, $\Delta$ and $\cap$ define a ring structure on the [[power set]] $\mathcal{P}(X)$ of a set $X$ (the set of subsets of $X$), in contrast to union and intersection. This ring is the same as the ring of $\mathbb{Z}/2\mathbb{Z}$-valued functions on $X$ (with [[pointwise multiplication]] and addition). Cf. also [[Boolean algebra]] and [[Boolean ring]] for the symmetric difference operation in an arbitrary Boolean algebra.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,  "Introduction to set theory and topology" , Pergamon  (1961)  pp. 34, 35  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,  "Introduction to set theory and topology" , Pergamon  (1961)  pp. 34, 35  (Translated from French)</TD></TR>
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</table>

Revision as of 22:12, 5 December 2014

2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

An operation on sets. Given two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by $$ A \Delta B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B) $$ where the symbols $\cup$, $\cap$, $\setminus$, ${}'$ denote the operations of union, intersection, difference, and complementation of sets, respectively.


Comments

The symmetric difference operation is associative, i.e. $A \Delta (B \Delta C) = (A \Delta B) \Delta C$, and intersection is distributive over it, i.e. $A \cap (B \Delta C) = (A \cap B) \Delta (A \cap C)$. Thus, $\Delta$ and $\cap$ define a ring structure on the power set $\mathcal{P}(X)$ of a set $X$ (the set of subsets of $X$), in contrast to union and intersection. This ring is the same as the ring of $\mathbb{Z}/2\mathbb{Z}$-valued functions on $X$ (with pointwise multiplication and addition). Cf. also Boolean algebra and Boolean ring for the symmetric difference operation in an arbitrary Boolean algebra.

References

[a1] C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 34, 35 (Translated from French)
How to Cite This Entry:
Symmetric difference of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_difference_of_sets&oldid=35382
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article