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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Symmetric difference ： ウィキペディア英語版 Symmetric difference In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by :$A\backslash ,\backslash triangle\backslash ,B,$ or :$A\; \backslash ominus\; B,$ or :$A\; \backslash oplus\; B.$ For example, the symmetric difference of the sets $\backslash$ and $\backslash$ is $\backslash$. The symmetric difference of the set of all students and the set of all females consists of all non-female students together with all female non-students. 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: :$A\backslash ,\backslash triangle\backslash ,B\; =\; (A\; \backslash smallsetminus\; B)\; \backslash cup\; (B\; \backslash smallsetminus\; A),\backslash ,$ The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation: :$A\backslash ,\backslash triangle\backslash ,B\; =\; \backslash .$ The same fact can be stated as the indicator function (which we denote here by $\backslash chi$) of the symmetric difference being the XOR (or addition mod 2) of the indicator functions of its two arguments: $\backslash chi\_\; =\; \backslash chi\_A\; \backslash oplus\; \backslash chi\_B$ or using the Iverson bracket notation $(\backslash in\; A\backslash ,\backslash triangle\backslash ,B)\; =\; (\backslash in\; A)\; \backslash oplus\; (\backslash in\; B)$. The symmetric difference can also be expressed as the union of the two sets, minus their intersection: :$A\backslash ,\backslash triangle\backslash ,B\; =\; (A\; \backslash cup\; B)\; \backslash smallsetminus\; (A\; \backslash cap\; B),$ In particular, $A\backslash triangle\; B\backslash subseteq\; A\backslash cup\; B$; the equality in this non-strict inclusion occurs if and only if $A$ and $B$ are disjoint sets. Furthermore, if we denote $D\; =\; A\backslash triangle\; B$ and $I\; =\; A\; \backslash cap\; B$, then $D$ and $I$ are always disjoint, so $D$ and $I$ partition $A\; \backslash cup\; B$. 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 :$A\backslash ,\backslash cup\backslash ,B\; =\; (A\backslash ,\backslash triangle\backslash ,B)\backslash ,\backslash triangle\backslash ,(A\; \backslash cap\; B)$. The symmetric difference is commutative and associative (and consequently the leftmost set of parentheses in the previous expression were thus redundant): :$A\backslash ,\backslash triangle\backslash ,B\; =\; B\backslash ,\backslash triangle\backslash ,A,\backslash ,$ :$(A\backslash ,\backslash triangle\backslash ,B)\backslash ,\backslash triangle\backslash ,C\; =\; A\backslash ,\backslash triangle\backslash ,(B\backslash ,\backslash triangle\backslash ,C).\backslash ,$ The empty set is neutral, and every set is its own inverse: :$A\backslash ,\backslash triangle\backslash ,\backslash varnothing\; =\; A,\backslash ,$ :$A\backslash ,\backslash triangle\backslash ,A\; =\; \backslash varnothing.\backslash ,$ Taken together, we see that the power set of any set ''X'' becomes an abelian group if we use the symmetric difference as 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 Z2. 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: :$(A\backslash ,\backslash triangle\backslash ,B)\backslash ,\backslash triangle\backslash ,(B\backslash ,\backslash triangle\backslash ,C)\; =\; A\backslash ,\backslash triangle\backslash ,C.\backslash ,$ 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''. (But note that for the diameter of the symmetric difference the triangle inequality does not hold.) Intersection distributes over symmetric difference: :$A\; \backslash cap\; (B\backslash ,\backslash triangle\backslash ,C)\; =\; (A\; \backslash cap\; B)\backslash ,\backslash triangle\backslash ,(A\; \backslash cap\; C),$ 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: * $A\backslash triangle\; B=A^c\backslash triangle\; B^c$, where $A^c$,$B^c$ is $A$'s complement,$B$'s complement, respectively, relative to any (fixed) set that contains both. * $\backslash left(\backslash bigcup\_\}B\_\backslash alpha\backslash right)\backslash subseteq\backslash bigcup\_$ is an arbitrary non-empty index set. * If $f\; :\; S\; \backslash rightarrow\; T$ is any function and $A,\; B\; \backslash subseteq\; T$ are any sets in $f$'s codomain, then $f^\backslash left(A\; \backslash Delta\; B\backslash right)\; =\; f^\backslash left(A\backslash right)\; \backslash Delta\; f^\backslash left(B\backslash right)$. The symmetric difference can be defined in any Boolean algebra, by writing :$x\backslash ,\backslash triangle\backslash ,y\; =\; (x\; \backslash lor\; y)\; \backslash land\; \backslash lnot(x\; \backslash land\; y)\; =\; (x\; \backslash land\; \backslash lnot\; y)\; \backslash lor\; (y\; \backslash land\; \backslash lnot\; x)\; =\; x\; \backslash oplus\; y.$ This operation has the same properties as the symmetric difference of sets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Symmetric difference」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.