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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 Bijection, injection and surjection ： ウィキペディア英語版 Bijection, injection and surjection In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which ''arguments'' (input expressions from the domain) and ''images'' (output expressions from the codomain) are related or ''mapped to'' each other. A function maps elements from its domain to elements in its codomain. Given a function $f:\; \backslash ;\; A\; \backslash to\; B$ *The function is injective (one-to-one) if every element of the codomain is mapped to by ''at most'' one element of the domain. An injective function is an injection. Notationally: :$\backslash forall\; x,\; y\; \backslash in\; A,\; f(x)\; =\; f(y)\; \backslash Rightarrow\; x\; =\; y.\backslash$ : Or, equivalently (using logical transposition), :$\backslash forall\; x,y\; \backslash in\; A,\; x\; \backslash neq\; y\; \backslash Rightarrow\; f(x)\; \backslash neq\; f(y).\backslash$ *The function is surjective (onto) if every element of the codomain is mapped to by ''at least'' one element of the domain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection. Notationally: :$\backslash forall\; y\; \backslash in\; B,\; \backslash exists\; x\; \backslash in\; A\; \backslash text\; y\; =\; f(x).\backslash$ *The function is bijective (one-to-one and onto or one-to-one correspondence) if every element of the codomain is mapped to by ''exactly'' one element of the domain. (That is, the function is ''both'' injective and surjective.) A bijective function is a bijection. An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with ''more than one'' argument). The four possible combinations of injective and surjective features are illustrated in the diagrams to the right. ==Injection== (詳細はiff for all $a,b\; \backslash in\; A$, we have $f(a)\; =\; f(b)\; \backslash Rarr\; a\; =\; b.$ *A function ''f'' : ''A'' → ''B'' is injective if and only if ''A'' is empty or ''f'' is left-invertible; that is, there is a function g : f(A) → A such that ''g'' o ''f'' = identity function on ''A''. Here f(A) is the image of f. *Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More precisely, every injection ''f'' : ''A'' → ''B'' can be factored as a bijection followed by an inclusion as follows. Let ''f''''R'' : ''A'' → ''f''(''A'') be ''f'' with codomain restricted to its image, and let ''i'' : ''f''(''A'') → ''B'' be the inclusion map from ''f''(''A'') into ''B''. Then ''f'' = ''i'' o ''f''''R''. A dual factorisation is given for surjections below. *The composition of two injections is again an injection, but if ''g'' o ''f'' is injective, then it can only be concluded that ''f'' is injective. See the figure at right. *Every embedding is injective. 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「Bijection, injection and surjection」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.