Symmetric difference of sets

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)