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Concept
Symmetric difference
Summary
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is .
The symmetric difference of the sets A and B is commonly denoted by or
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. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
The symmetric difference is equivalent to the union of both relative complements, that is:
The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation:
The same fact can be stated as the indicator function (denoted here by ) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: or using the Iverson bracket notation .
The symmetric difference can also be expressed as the union of the two sets, minus their intersection:
In particular, ; the equality in this non-strict inclusion occurs if and only if and are disjoint sets. Furthermore, denoting and , then and are always disjoint, so and partition . 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
The symmetric difference is commutative and associative:
The empty set is neutral, and every set is its own inverse:
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 2) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups.
Official source
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