Difference between revisions of "Symmetric difference of sets"
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| − | + | {{TEX|done}}{{MSC|03E}} | |
| − | + | An operation on sets. Given two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by | |
| − | + | $$ | |
| − | where the symbols | + | A \Delta B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B) |
| + | $$ | ||
| + | where the symbols $\cup$, $\cap$, $\setminus$, ${}'$ denote the operations of [[union 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. | + | 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> | + | <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> | ||
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
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