Symmetric difference
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 formula_1 and formula_2 is formula_3. The symmetric difference of the sets "A" and "B" is commonly denoted by formula_4 or formula_5 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. Properties. 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 formula_8) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: formula_9 or using  the Iverson bracket notation formula_10. The symmetric difference can also be expressed as the union of the two sets, minus their intersection: In particular, formula_12; the equality in this non-strict inclusion occurs if and only if formula_13 and formula_14 are disjoint sets. Furthermore, denoting formula_15 and formula_16, then formula_17 and formula_18 are always disjoint, so formula_17 and formula_18 partition formula_21. 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. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set. 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 2-group. Consequently, the group induced by the symmetric difference is in fact a vector space over the field with 2 elements Z. If "X" is finite, then the singletons form a basis of this vector space, and its 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 join of the two multisets, where for each double set both can be removed. In particular: This implies triangle inequality: 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 distributes over symmetric difference: and this shows that the power set of "X" becomes a 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: The symmetric difference can be defined in any Boolean algebra, by writing This operation has the same properties as the symmetric difference of sets. Section::::"n"-ary symmetric difference. The repeated symmetric difference is in a sense equivalent to an operation on a multiset of sets giving the set of elements which are in an odd number of sets. As above, the symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection: formula_41 Evidently, this is well-defined only when each element of the union formula_42 is contributed  by a finite number of elements of formula_43. Suppose formula_44 is a multiset and formula_45. Then there is a formula for formula_46, the number of elements in formula_47, given solely in terms of intersections of elements of formula_43: formula_49 Symmetric difference on measure spaces. As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. First consider a finite set "S" and the counting measure on subsets given by their size. Now consider two subsets of "S" and set their distance  apart as the size of their symmetric difference. This distance is in fact a metric, which makes the power set on "S" a metric space. If "S" has "n" elements, then the distance from the empty set to "S" is "n", and this is the maximum distance for any pair of subsets. Using the ideas of measure theory, the separation of measurable sets can be defined to be the measure of their symmetric difference. If μ is a σ-finite measure defined on a σ-algebra Σ, the function is a pseudometric on Σ. "d" becomes a metric if Σ  is considered modulo the equivalence relation "X" ~ "Y" if and only if formula_51. It is sometimes called Fréchet-Nikodym metric. The resulting metric space is separable if and only if L(μ) is separable. If formula_52, we have: formula_53. Indeed, If formula_55 is a measure space and formula_56 are measurable sets, then their symmetric difference is also measurable: formula_57. One may define an equivalence relation on measurable sets by letting formula_58 and formula_59 be related if formula_60. This relation is denoted formula_61. Given formula_62, one writes formula_63 if to each formula_64 there's some formula_65 such that formula_66. The relation  formula_67 is a partial order on the family of subsets of formula_68. We write formula_69 if formula_63 and formula_71. The relation "formula_72" is an equivalence relationship between the subsets of formula_68. The "symmetric closure" of formula_74 is the collection of all formula_68-measurable sets that are formula_72 to some formula_77. The symmetric closure of formula_74 contains formula_74. If formula_74 is a sub-formula_81-algebra of formula_68, so is the symmetric closure of formula_74. formula_61 iff formula_85 formula_86 almost everywhere. Hausdorff distance vs. symmetric difference. The Hausdorff distance and the (area of the) symmetric difference are both pseudo-metrics on the  set of measurable geometric shapes. However, they behave quite differently. The figure at the right shows two sequences of shapes, "Red" and "Red ∪ Green". When the Hausdorff distance between them becomes smaller, the area of the symmetric difference between them becomes larger, and vice versa. By continuing these sequences in both directions, it is possible to get two sequences such that the Hausdorff distance between them converges to 0 and the symmetric distance between them diverges, or vice versa.