[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/113090#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/113090","headline":"\u5168\u5c04 – Wikipedia","name":"\u5168\u5c04 – Wikipedia","description":"\u51fa\u5178\u306f\u5217\u6319\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u811a\u6ce8\u306a\u3069\u3092\u7528\u3044\u3066\u3069\u306e\u8a18\u8ff0\u306e\u60c5\u5831\u6e90\u3067\u3042\u308b\u304b\u3092\u660e\u8a18\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u3092\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\uff082021\u5e7412\u6708\uff09 \u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u5199\u50cf\u304c\u5168\u5c04\u7684\uff08\u305c\u3093\u3057\u3083\u3066\u304d\u3001\u82f1: surjective, onto\uff09\u3067\u3042\u308b\u3068\u306f\u3001\u305d\u306e\u7d42\u57df\u3068\u306a\u308b\u96c6\u5408\u306e\u5143\u306f\u3069\u308c\u3082\u305d\u306e\u5199\u50cf\u306e\u50cf\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u3053\u3068\u3092\u8a00\u3046\u3002\u5373\u3061\u3001\u96c6\u5408 X \u304b\u3089\u96c6\u5408 Y \u3078\u306e\u5199\u50cf f \u306b\u3064\u3044\u3066\u3001Y \u306e\u5404\u5143 y \u306b\u5bfe\u3057 f(x) = y \u3068\u306a\u308b\u3088\u3046\u306a X \u306e\u5143 x","datePublished":"2022-03-17","dateModified":"2022-03-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b4\/Ambox_important.svg\/40px-Ambox_important.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b4\/Ambox_important.svg\/40px-Ambox_important.svg.png","height":"40","width":"40"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/113090","about":["Wiki"],"wordCount":7407,"articleBody":"\u51fa\u5178\u306f\u5217\u6319\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u811a\u6ce8\u306a\u3069\u3092\u7528\u3044\u3066\u3069\u306e\u8a18\u8ff0\u306e\u60c5\u5831\u6e90\u3067\u3042\u308b\u304b\u3092\u660e\u8a18\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u3092\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\uff082021\u5e7412\u6708\uff09\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u5199\u50cf\u304c\u5168\u5c04\u7684\uff08\u305c\u3093\u3057\u3083\u3066\u304d\u3001\u82f1: surjective, onto\uff09\u3067\u3042\u308b\u3068\u306f\u3001\u305d\u306e\u7d42\u57df\u3068\u306a\u308b\u96c6\u5408\u306e\u5143\u306f\u3069\u308c\u3082\u305d\u306e\u5199\u50cf\u306e\u50cf\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u3053\u3068\u3092\u8a00\u3046\u3002\u5373\u3061\u3001\u96c6\u5408 X \u304b\u3089\u96c6\u5408 Y \u3078\u306e\u5199\u50cf f \u306b\u3064\u3044\u3066\u3001Y \u306e\u5404\u5143 y \u306b\u5bfe\u3057 f(x) = y \u3068\u306a\u308b\u3088\u3046\u306a X \u306e\u5143 x \u304c\uff08\u4e00\u822c\u306b\u306f\u8907\u6570\u3042\u3063\u3066\u3082\u3088\u3044\u304c\uff09\u5bfe\u5fdc\u3055\u305b\u3089\u308c\u308b\u3068\u304d\u3001\u5199\u50cf f \u306f\u5168\u5c04 (surjection, onto mapping\/function) \u3067\u3042\u308b\u3068\u3044\u3046\u3002\u5168\u5199\uff08\u3042\u308b\u3044\u306f\u5168\u5199\u50cf\uff09\u3068\u3082\u66f8\u304f\u3002  \u57df X\uff08\u8d64\uff09\u304b\u3089\u4f59\u57df Y\uff08\u9752+\u9ec4\uff09\u3078\u306e\u5199\u50cf f \u306e\u6a21\u5f0f\u56f3\uff08\u4f59\u57df Y \u306e\u5185\u5074\u306e\u5c0f\u3055\u3044\u6955\u5186\uff08\u9ec4\uff09\u306f f \u306e\u5024\u57df\uff09\u3002\u3053\u308c\u306f\u4e00\u822c\u306b\u306f\u5168\u5c04\u3092\u8868\u3057\u3066\u3044\u306a\u3044\uff08\u4e00\u3064\u3082\u9752\u306b\u5857\u3089\u308c\u308b\u70b9\u304c\u306a\u3044\u3068\u304d\u306e\u307f\u5168\u5c04\uff09\u3002\u5168\u5c04\uff08\u304a\u3088\u3073\u5358\u5c04\u3001\u53cc\u5c04\uff09\u306e\u8a9e\u306f20\u4e16\u7d00\u30d5\u30e9\u30f3\u30b9\u306e\u6570\u5b66\u7d50\u793e\u30d6\u30eb\u30d0\u30ad\uff081935\u5e74\u4ee5\u964d\u300e\u6570\u5b66\u539f\u8ad6\u300f\u30b7\u30ea\u30fc\u30ba\u3092\u520a\u884c\u3057\u3066\u3044\u308b\uff09\u306b\u3088\u308a\u5c0e\u5165\u3055\u308c\u305f\u3082\u306e\u3067\u3042\u308b\u3002\u63a5\u982d\u8f9e sur- \u306f\u30d5\u30e9\u30f3\u30b9\u8a9e\u3067\u300c\u4e0a\u306e\u300d\u3092\u610f\u5473\u3057\u3001\u5199\u50cf\u306e\u59cb\u57df\u304c\u7d42\u57df\u5168\u4f53\u3092\u3059\u3063\u307d\u308a\u8986\u3044\u5c3d\u304f\u3059\u3088\u3046\u306b\u5199\u3057\u8fbc\u307e\u308c\u308b\u30a4\u30e1\u30fc\u30b8\u3092\u53cd\u6620\u3057\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u308b\u3002sur, in, bi, jection \u3044\u305a\u308c\u3082\u30e9\u30c6\u30f3\u8a9e\u6e90\u3067\u3042\u308b\u3002\u5199\u50cf f: A \u2192 B \u306b\u3064\u3044\u3066\u3001f \u306e\u5024\u57df f(A) := {f(a) | a \u2208 A} \u304c\u7d42\u57df\uff08\u4f59\u57df\uff09B \u3068\u4e00\u81f4\u3059\u308b\uff08\u3064\u307e\u308a f(A) = B\uff09\u306a\u3089\u3070\u3001\u5199\u50cf f: A \u2192 B \u306f \u5168\u5c04 (surjection) \u3067\u3042\u308b\u3068\u3044\u3046\u3002f \u306f\u4f59\u57df B \u3078\u306e\u5168\u5c04\u7684 (surjective) \u306a\u5199\u50cf\u3067\u3042\u308b\u3001B \u306e\u4e0a\u3078\u306e (onto) \u5199\u50cf\u3067\u3042\u308b\u306a\u3069\u3068\u3082\u3044\u3046[\u6ce8\u91c8 1]\u3002\u8a18\u53f7\u3067\u66f8\u3051\u3070\u3001f: A \u2192 B \u304c\u5168\u5c04\u3067\u3042\u308b\u3068\u306f \u2200b \u2208 B, \u2203a \u2208 A, f(a) = b \u3092\u6e80\u8db3\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u3057\u3070\u3057\u3070\u93c3\u304c\u4e8c\u3064\u306e\u77e2\u5370\u3092\u4f7f\u3063\u3066 f:A\u21a0B{displaystyle fcolon Atwoheadrightarrow B} \u3068\u8868\u3059\u3002\u5168\u5c04\u3067\u3042\u308a\u5358\u5c04\u3067\u306a\u3044\u5358\u5c04\u3067\u3042\u308a\u5168\u5c04\u3067\u306a\u3044\u5168\u5c04\u3067\u3082\u5358\u5c04\u3067\u3082\u306a\u3044\u5b9f\u6570 x \u306b\u5bfe\u3057\u3001\u305d\u306e\u81ea\u4e57 x2 \u3092\u5bfe\u5fdc\u3055\u305b\u308b\u975e\u8ca0\u5b9f\u6570\u5024\u5199\u50cf f:R\u220bx\u21a6x2\u2208R+{displaystyle fcolon mathbb {R} ni xmapsto x^{2}in mathbb {R} _{+}} \u306f\u5168\u5c04\u3067\u3042\u308b\u3002\u305f\u3060\u3057 R+ \u3067\u975e\u8ca0\u5b9f\u6570\u5168\u4f53\u306e\u96c6\u5408\u3092\u8868\u3057\u3066\u3044\u308b\u3002\u5b9f\u969b\u3001x \u2265 0 \u306b\u5bfe\u3057\u3001\u305d\u306e\u975e\u8ca0\u5e73\u65b9\u6839 \u221ax \u3092\u3068\u308c\u3070\u3001f(\u221ax) = x \u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\uff08\u8ca0\u306e\u5e73\u65b9\u6839 \u2212\u221ax \u3092\u3068\u3063\u3066\u3082\u69cb\u308f\u306a\u3044\uff09\u3002\u7d42\u57df\u3092\u5909\u66f4\u3057\u3066\u3001\u5358\u306b\u5b9f\u95a2\u6570 f:R\u220bx\u21a6x2\u2208R{displaystyle fcolon mathbb {R} ni xmapsto x^{2}in mathbb {R} } \u3068\u8003\u3048\u305f\u306e\u3067\u306f\u5168\u5c04\u306b\u306f\u306a\u3089\u306a\u3044\u3002\u81ea\u4e57\u3057\u3066\u8ca0\u306b\u306a\u308b\u5b9f\u6570\u306f\u5b58\u5728\u3057\u306a\u3044\u304b\u3089\u3067\u3042\u308b\u3002\u4efb\u610f\u306e\u96c6\u5408 X \u306b\u304a\u3044\u3066\u3001X \u4e0a\u306e\u6052\u7b49\u5909\u63db idX:X\u220bx\u21a6x\u2208X{displaystyle {text{id}}_{X}colon Xni xmapsto xin X} \u306f\u5168\u5c04\uff08\u5b9f\u306f\u53cc\u5c04\uff09\u3067\u3042\u308b\u3002\u30c7\u30ab\u30eb\u30c8\u7a4d A \u00d7 B \u306e\u5404\u6210\u5206\u3078\u306e\u5c04\u5f71 pA:A\u00d7B\u220b(a,b)\u21a6a\u2208A,\u00a0pB:A\u00d7B\u220b(a,b)\u21a6b\u2208B{displaystyle p_{A}colon Atimes Bni (a,b)mapsto ain A, p_{B}colon Atimes Bni (a,b)mapsto bin B}\u306f\u5168\u5c04\u3067\u3042\u308b\u3002\u5b9f2\u6b21\u591a\u9805\u5f0f\u5168\u4f53 R2[x]:={ax2+bx+c\u00a0;\u00a0(a,b,c)\u2208(R\u2216{0})\u00d7R\u00d7R}{displaystyle mathbb {R} _{2}[x]:={ax^{2}+bx+c ; (a,b,c)in (mathbb {R} setminus {0})times mathbb {R} times mathbb {R} }} \u304b\u3089 R{displaystyle mathbb {R} } \u3078\u306e\u5199\u50cf D \u3092 D(ax2+bx+c):=b2\u22124ac{displaystyle D(ax^{2}+bx+c):=b^{2}-4ac} \u3068\u5b9a\u7fa9\u3059\u308b\u3068\u3001D \u306f\u5168\u5c04\u3067\u3042\u308b\uff08\u4efb\u610f\u306e r\u2208R{displaystyle rin mathbb {R} } \u306b\u5bfe\u3057\u3066\u3001\u4f8b\u3048\u3070 x2\u2212r\/4\u21a6r{displaystyle x^{2}-r\/4mapsto r} \u3067\u3042\u308b\uff09\u3002\u6b63\u306e\u6574\u6570 n \u306b\u5bfe\u3057\u3001det:Mn(R)\u220bA\u21a6detA\u2208R{displaystyle det colon M_{n}(mathbb {R} )ni Amapsto det Ain mathbb {R} }\u306f\u5168\u5c04\u3067\u3042\u308b\uff08\u3053\u3053\u3067 Mn(R){displaystyle M_{n}(mathbb {R} )}\u306f\u5b9f n \u6b21\u6b63\u65b9\u884c\u5217\u5168\u4f53\u3067\u3042\u308a\uff0c detA{displaystyle det A}\u306f\u884c\u5217\u5f0f\u3092\u8868\u3059\uff09\uff0e\u5b9f\u969b\uff0c\u4efb\u610f\u306e\u5b9f\u6570 r \u306b\u5bfe\u3057\u3066\u3001\u5bfe\u89d2\u884c\u5217 diag(r, 1, \u2026, 1) \u306e\u884c\u5217\u5f0f\u306f r \u3067\u3042\u308b\u3002\u6307\u6570\u95a2\u6570exp:R\u220bx\u21a6ex\u2208(0,\u221e){displaystyle exp colon mathbb {R} ni xmapsto e^{x}in (0,infty )} \u306f\u5168\u5c04\u3067\u3042\u308b\uff0e\u5b9f\u969b\uff0c\u4efb\u610f\u306ey\u2208(0,\u221e){displaystyle yin (0,infty )} \u306b\u5bfe\u3057\u3066\uff0cx:=log\u2061y{displaystyle x:=log y} \u3068\u3068\u308c\u3070exp\u2061(x)=ex=y{displaystyle exp(x)=e^{x}=y}\u3067\u3042\u308b\uff0e\u8907\u7d20\u6570\u306b\u5bfe\u3057\u3066\u305d\u306e\u5b9f\u90e8\uff0c\u865a\u90e8\uff0c\u7d76\u5bfe\u5024\u3092\u4e0e\u3048\u308b\u5199\u50cf\u211c:C\u220bz\u21a6(z+z\u00af)\/2\u2208R,\u00a0\u2111:C\u220bz\u21a6(z\u2212z\u00af)\/2i\u2208R,\u00a0|\u22c5|:C\u220bz\u21a6\u211c(z)2+\u2111(z)2\u2208R{displaystyle Re colon mathbb {C} ni zmapsto (z+{overline {z}})\/2in mathbb {R} , Im colon mathbb {C} ni zmapsto (z-{overline {z}})\/2iin mathbb {R} , |cdot |colon mathbb {C} ni zmapsto {sqrt {Re (z)^{2}+Im (z)^{2}}}in mathbb {R} }\u306f\u3044\u305a\u308c\u3082\u5168\u5c04\u3067\u3042\u308b\uff0e  \u5e73\u9762\u4e0a\u306b\u8868\u3057\u305f\u5168\u5c04\u306e\u6a21\u5f0f\u56f3\u3002\u51fd\u6570 f: X \u2192 Y; y = f(x)\uff08X = \u51fd\u6570\u306e\u5b9a\u7fa9\u57df, Y = \u51fd\u6570\u306e\u5024\u57df\uff09\u3002\u5024\u57df\u306e\u5404\u5143\u306e\u4e0a\u306b\u5b9a\u7fa9\u57df\u306e\u5143\u304c\u898f\u5247 f \u306b\u5f93\u3063\u3066\u5199\u3055\u308c\u308b\uff08\u5b9a\u7fa9\u57df\u306e\u8907\u6570\u306e\u5143\u304c\u5024\u57df\u306e\u540c\u3058\u5143\u306b\u5199\u3063\u3066\u3082\u3088\u3044\uff09\u3002\u5de6\uff1a f \u304c\u5168\u5c04\u306b\u306a\u308b\u3088\u3046\u306a\u5b9a\u7fa9\u57df\u306e\u4e00\u3064\u3002\u53f3\uff1a \u4e8c\u3064\u306e\u5b9a\u7fa9\u57df X1, X2 \u304c\u793a\u3055\u308c\u3066\u3044\u308b\u304c\u3001\u4f55\u308c\u306e\u5834\u5408\u3082 f \u306f\u5168\u5c04\u306b\u306a\u308b\u3002  \u5e73\u9762\u4e0a\u306b\u8868\u3057\u305f\u5168\u5c04\u3067\u306a\u3044\u5834\u5408\u306e\u6a21\u5f0f\u56f3\u3002\u4f59\u57df Y \u306e\u4e00\u90e8\u306e\u5143 y \u306f\u9069\u5f53\u306a x \u2208 X \u3092\u3068\u3063\u3066 y = f(x) \u3068\u66f8\u3051\u308b\u304c\u3001\u305d\u3046\u306f\u66f8\u3051\u306a\u3044\u90e8\u5206\u3082\u3042\u308b\u3002\u5de6\uff1a y0 \u306f Y \u306b\u5c5e\u3059\u304c\u3001y0 = f(x0) \u3068\u306a\u308b x0 \u2208 X \u304c\u306a\u3044\u3002\u53f3\uff1a y1, y2, \u304a\u3088\u3073 y3 \u306f Y \u306e\u5143\u3060\u304c\u3001y1 = f(x1), y2 = f(x2), \u304a\u3088\u3073 y3 = f(x3) \u3068\u306a\u308b\u3088\u3046\u306a x1, x2, \u304a\u3088\u3073 x3 \u306f X \u306e\u4e2d\u306b\u306f\u7121\u3044\u3002\u5199\u50cf\u304c\u53cc\u5c04\uff08\u5168\u5358\u5c04\uff09\u3068\u306a\u308b\u306e\u306f\u3001\u305d\u308c\u304c\u5358\u5c04\u304b\u3064\u5168\u5c04\u3068\u306a\u308b\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308b\u3002\u51fd\u6570\u3092\uff08\u3088\u304f\u3084\u308b\u3088\u3046\u306b\uff09\u305d\u306e\u30b0\u30e9\u30d5\u3068\u540c\u4e00\u8996\u3057\u3066\u8003\u3048\u308b\u3068\u304d\u3001\u5358\u5c04\u6027\u3068\u306f\u7570\u306a\u308a\u3001\u5168\u5c04\u6027\u3092\u51fd\u6570\u306e\u30b0\u30e9\u30d5\u306e\u307f\u304b\u3089\u8aad\u307f\u53d6\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002\u5168\u5c04\u6027\u306f\u51fd\u6570\u81ea\u4f53\u306e\u6027\u8cea\u3068\u3044\u3046\u3088\u308a\u306f\u51fd\u6570\u3068\u4f59\u57df\u3068\u306e\u95a2\u4fc2\u6027\u3068\u898b\u308b\u3079\u304d\u3082\u306e\u3067\u3042\u308b\u3002Table of Contents\u53f3\u53ef\u9006\u6027[\u7de8\u96c6]\u5168\u578b\u5c04\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u4e8c\u9805\u95a2\u4fc2\u3068\u3057\u3066\u306e\u5168\u5c04[\u7de8\u96c6]\u5168\u5c04\u306e\u59cb\u57df\u306e\u6fc3\u5ea6[\u7de8\u96c6]\u5408\u6210\u3068\u5206\u89e3[\u7de8\u96c6]\u8a98\u5c0e\u3055\u308c\u305f\u5168\u5c04\u30fb\u53cc\u5c04[\u7de8\u96c6]\u6570\u3048\u4e0a\u3052[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u53f3\u53ef\u9006\u6027[\u7de8\u96c6]\u5199\u50cf g: Y \u2192 X \u304c\u5199\u50cf f: X \u2192 Y \u306e\u53f3\u9006\u5199\u50cf\u3067\u3042\u308b\u3068\u306f\u3001f(g(y)) = y\uff08\u3064\u307e\u308a g \u306e\u52b9\u679c\u304c f \u306b\u3088\u3063\u3066\u6253\u3061\u6d88\u3055\u308c\u308b\uff09\u304c Y \u306e\u5404\u5143 y \u3067\u6210\u308a\u7acb\u3064\u3068\u304d\u306b\u8a00\u3046\u3002\u8a00\u8449\u3092\u5909\u3048\u308c\u3070\u3001g \u3068 f \u3068\u306e\u3053\u306e\u9806\u756a\u3067\u306e\u5408\u6210 f \u2218 g \u304c g \u306e\u5b9a\u7fa9\u57df Y \u4e0a\u306e\u6052\u7b49\u5199\u50cf idY \u3068\u306a\u308b\u3068\u304d\u3001g \u304c f \u306e\u53f3\u9006\u3067\u3042\u308b\u3068\u3044\u3046\u3002\u9006\u9806\u306e g \u2218 f \u304c f \u306e\u5b9a\u7fa9\u57df X \u4e0a\u306e\u6052\u7b49\u5199\u50cf\u3067\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304b\u3089\u3001\u5199\u50cf g \u306f\u5fc5\u305a\u3057\u3082 f \u306e\uff08\u5b8c\u5168\uff09\u9006\u5199\u50cf\u3067\u3042\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\u5373\u3061\u3001f \u306f g \u3092\u6253\u3061\u6d88\u3059\u304c\u3001\u9006\u306f\u5fc5\u305a\u3057\u3082\u6210\u308a\u7acb\u305f\u306a\u3044\u3002  \u53f3\u9006\u3092\u6301\u3064\u4efb\u610f\u306e\u5199\u50cf\u306f\u5168\u5c04\u3067\u3042\u308b\u304c\u3001\u300c\u4efb\u610f\u306e\u5168\u5c04\u304c\u53f3\u9006\u5199\u50cf\u3092\u6301\u3064\u300d\u3068\u3044\u3046\u547d\u984c\u306f\u9078\u629e\u516c\u7406\u306b\u540c\u5024\u3067\u3042\u308b\u3002f: X \u2192 Y \u304c\u5168\u5c04\u3067 B \u304c Y \u306e\u90e8\u5206\u96c6\u5408\u3067\u3042\u308b\u3068\u304d\u3001f(f \u22121(B)) = B \u304c\u6210\u308a\u7acb\u3064\u3002\u3064\u307e\u308a B \u306f\u305d\u306e\u539f\u50cf f \u22121(B) \u304b\u3089\u56de\u5fa9\u3055\u308c\u308b\u3002\u5168\u578b\u5c04\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]  \u5199\u50cf f: X \u2192 Y \u304c\u5168\u5c04\u3068\u306a\u308b\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f\u3001\u305d\u308c\u304c\u53f3\u6d88\u7d04\u7684\u3067\u3042\u308b\u3053\u3068[1]\u3001\u5373\u3061\u300c\u4e0e\u3048\u3089\u308c\u305f\u5199\u50cf g1, g2: Y \u2192 Z \u304c g1 \u2218 f = g2 \u2218 f \u3092\u6e80\u305f\u3059\u9650\u308a\u5e38\u306b g1 = g2 \u304c\u8a00\u3048\u308b\u3053\u3068\u300d\u3067\u3042\u308b\u3002\u3053\u306e\u6027\u8cea\u306f\u3001\u5199\u50cf\u3068\u305d\u306e\u5408\u6210\u306b\u3088\u3063\u3066\u5b9a\u5f0f\u5316\u3055\u308c\u3066\u3044\u308b\u304b\u3089\u3001\u3088\u308a\u4e00\u822c\u306b\u570f\u306b\u304a\u3051\u308b\u5c04 (morphism) \u3068\u305d\u306e\u5408\u6210\u306b\u3064\u3044\u3066\u306e\u6027\u8cea\u306b\u4e00\u822c\u5316\u3067\u304d\u308b\u3002\u5373\u3061\u3001\u53f3\u6d88\u7d04\u7684\u306a\u5c04\u306f\u30a8\u30d4\u5c04\u3042\u308b\u3044\u306f\u5168\u578b\u5c04\uff08\u570f\u8ad6\u7684\u5168\u5c04\uff09\u3067\u3042\u308b\u3068\u3044\u3046\u3002\u5199\u50cf\u304c\uff08\u96c6\u5408\u8ad6\u7684\uff09\u5168\u5c04 (surjection) \u306a\u3089\u3070\u3001\u305d\u308c\u306f\u3061\u3087\u3046\u3069\u96c6\u5408\u306e\u570f\u306b\u304a\u3051\u308b\u5168\u578b\u5c04 (epimorphism) \u306b\u306a\u3063\u3066\u3044\u308b\u3002\u63a5\u982d\u8f9e\u306e \u1f10\u03c0\u03af \u306f\u30ae\u30ea\u30b7\u30e3\u8a9e\u3067\u300c\u4e0a\u306e\u300d\u3092\u610f\u5473\u3059\u308b\u8a00\u8449\u3067\u3042\u308b\u3002\u53f3\u9006\u578b\u5c04\u3092\u3082\u3064\u4efb\u610f\u306e\u5c04\u306f\u5168\u578b\u5c04\u3067\u3042\u308b\u304c\u3001\u9006\u306f\u4e00\u822c\u306b\u306f\u6b63\u3057\u304f\u306a\u3044\u3002\u5c04 f \u306e\u53f3\u9006 g \u306f f \u306b\u5bfe\u3059\u308b\u5207\u65ad\uff08\u82f1\u8a9e\u7248\uff09\u3068\u547c\u3070\u308c\u3001\u53f3\u9006\u3092\u6301\u3064\u5c04\u306f\u5206\u88c2\u578b\u5168\u578b\u5c04 (split epimorphism) \u3067\u3042\u308b\u3068\u3044\u3046\u3002  \u4e8c\u9805\u95a2\u4fc2\u3068\u3057\u3066\u306e\u5168\u5c04[\u7de8\u96c6]\u57df X \u304a\u3088\u3073\u4f59\u57df Y \u3092\u6301\u3064\u4efb\u610f\u306e\u5199\u50cf\u306f\uff08\u305d\u306e\u30b0\u30e9\u30d5\u3068\u540c\u4e00\u8996\u3059\u308b\u3053\u3068\u306b\u3088\u308a\uff09\u3001X \u3068 Y \u3068\u306e\u9593\u306e\u5de6\u5168\u57df\u7684\u304b\u3064\u53f3\u4e00\u610f\u306a\u4e8c\u9805\u95a2\u4fc2\u3068\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5f93\u3063\u3066\u3001\u57df X, \u4f59\u57df Y \u3092\u3082\u3064\u5168\u5c04\u306f\u3001X \u3068 Y \u3068\u306e\u9593\u306e\u5de6\u5168\u57df\u7684\u3001\u53f3\u4e00\u610f\u304b\u3064\u53f3\u5168\u57df\u7684\u306a\u4e8c\u9805\u95a2\u4fc2\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002\u5168\u5c04\u306e\u59cb\u57df\u306e\u6fc3\u5ea6[\u7de8\u96c6]\u5168\u5c04\u306e\u59cb\u57df\u306e\u6fc3\u5ea6\u306f\u3001\u4f59\u57df\u306e\u6fc3\u5ea6\u4ee5\u4e0a\u3067\u3042\u308b\u3002\u3064\u307e\u308a f: X \u2192 Y \u304c\u5168\u5c04\u306a\u3089\u3070\u3001X \u306f\u5c11\u306a\u304f\u3068\u3082 Y \u306e\u5143\u306e\uff08\u6fc3\u5ea6\u306e\u610f\u5473\u3067\u306e\uff09\u500b\u6570\u3068\u7b49\u3057\u3044\u6570\u306e\u5143\u3092\u542b\u3080\u3002\u305f\u3060\u3057\u3001\u3053\u306e\u3053\u3068\u306e\u8a3c\u660e\u306b\u306f\u3001Y \u306e\u4efb\u610f\u306e\u5143 y \u306b\u5bfe\u3057\u3066 f(g(y)) = y \u3092\u6e80\u305f\u3059\u5199\u50cf g: Y \u2192 X \u306e\u5b58\u5728\u3092\u8a00\u3046\u305f\u3081\u306b\u9078\u629e\u516c\u7406\u304c\u5fc5\u8981\u306b\u306a\u308b\u3002g \u304c\u5358\u5c04\u3067\u3042\u308b\u3053\u3068\u3092\u898b\u308b\u306e\u306f\u5bb9\u6613\u3067\u3042\u308b\u304b\u3089\u3001\u5b9a\u7fa9\u306b\u3088\u308a |Y| \u2264 |X| \u304c\u5f97\u3089\u308c\u308b\u3002\u7279\u306b\u3001X \u3068 Y \u304c\u540c\u3058\u6570\u306e\u5143\u3092\u6301\u3064\u6709\u9650\u96c6\u5408\u3067\u3042\u308b\u3068\u304d\u306b\u306f\u3001f: X \u2192 Y \u304c\u5168\u5c04\u3067\u3042\u308b\u3053\u3068\u3068 f \u304c\u5358\u5c04\u3067\u3042\u308b\u3053\u3068\u3068\u304c\u540c\u5024\u306b\u306a\u308b\u3002\u5408\u6210\u3068\u5206\u89e3[\u7de8\u96c6]    \u5408\u6210\u304c\u5168\u5c04\uff08\u5148\u306b\u65bd\u3059\u5199\u50cf\u304c\u5168\u5c04\u3067\u306a\u304f\u3068\u3082\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b\uff09\u5168\u5c04\u540c\u58eb\u306e\u5408\u6210\u306f\u5e38\u306b\u5168\u5c04\u3067\u3042\u308b\u3002\u5373\u3061\u3001f \u304a\u3088\u3073 g \u304c\u3068\u3082\u306b\u5168\u5c04\u3067\u3001g \u306e\u4f59\u57df\u304c f \u306e\u5b9a\u7fa9\u57df\u3068\u7b49\u3057\u3044\u3068\u304d\u3001\u5408\u6210\u5199\u50cf f \u2218 g \u306f\u5168\u5c04\u306b\u306a\u308b\u3002\u9006\u306b\u3001\u5408\u6210 f \u2218 g \u304c\u5168\u5c04\u306a\u3089\u3070 f \u306f\u5168\u5c04\uff08\u3060\u304c\u5148\u306b\u65bd\u3059\u307b\u3046\u306e g \u306f\u5fc5\u305a\u3057\u3082\u5168\u5c04\u3067\u306a\u304f\u3066\u3088\u3044\uff09\u3002\u3053\u306e\u6027\u8cea\u306f\u3001\u96c6\u5408\u306e\u570f\u306b\u304a\u3051\u308b\u5168\u5c04\u304b\u3089\u4efb\u610f\u306e\u570f\u306b\u304a\u3051\u308b\u4efb\u610f\u306e\u5168\u5c04\u306b\u4e00\u822c\u5316\u3055\u308c\u308b\u3002\u4efb\u610f\u306e\u5199\u50cf\u306f\u3001\u5168\u5c04\u3068\u5358\u5c04\u3068\u306e\u5408\u6210\u306e\u5f62\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5373\u3061\u3001h: X \u2192 Z \u3092\u4efb\u610f\u306e\u5199\u50cf\u3068\u3059\u308c\u3070\u3001\u5168\u5c04 f: X \u2192 Y \u3068\u5358\u5c04 g: Y \u2192 Z \u3067 h = g \u2218 f \u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u5b58\u5728\u3059\u308b\u3002\u3053\u308c\u3092\u898b\u308b\u306b\u306f\u3001\u96c6\u5408 Y \u306f X \u306e\u90e8\u5206\u96c6\u5408\u65cf Y:={h\u22121(z)={x\u2208X\u2223h(x)=z}\u2223z\u2208h(X)}{textstyle Y:={h^{-1}(z)={xin Xmid h(x)=z}mid zin h(X)}} \u3068\u3057\u3066\u5b9a\u3081\u308c\u3070\u3088\u3044\u3002\u3053\u3053\u306b\u73fe\u308c\u305f\u539f\u50cf\u306f\u4e92\u3044\u306b\u4ea4\u308f\u308a\u3092\u6301\u305f\u305a\u3001X \u306e\u5206\u5272\u3092\u4e0e\u3048\u308b\u3002\u3053\u306e\u3068\u304d\u3001f \u3068\u3057\u3066\u5404\u5143 x \u2208 X \u3092 x \u3092\u542b\u3080 Y \u306e\u5143\u3078\u5199\u3059\u5199\u50cf f:x\u21a6{x\u2032\u2208X\u2223h(x\u2032)=h(x)}{textstyle fcolon xmapsto {x’in Xmid h(x’)=h(x)}} \u3092\u3068\u308a\u3001g \u3068\u3057\u3066 Y \u306e\u5404\u5143\u304c\u542b\u3080 X \u306e\u5143\u304c h \u306b\u3088\u3063\u3066\u5199\u3055\u308c\u308b\u3068\u3053\u308d\u306e Z \u306e\u5143\u3078\u5199\u3059\u5199\u50cf g:h\u22121(z)\u21a6z{textstyle gcolon h^{-1}(z)mapsto z} \u3068\u3059\u308c\u3070\u3001f \u306f\u5c04\u5f71\u3086\u3048\u5168\u5c04\u3067\u3001g \u306f\u4f5c\u308a\u65b9\u304b\u3089\u5358\u5c04\u3068\u306a\u308a\u3001h = g \u2218 f \u304c\u6210\u308a\u7acb\u3064\u3002\u8a98\u5c0e\u3055\u308c\u305f\u5168\u5c04\u30fb\u53cc\u5c04[\u7de8\u96c6]\u4efb\u610f\u306e\u5199\u50cf\u306f\u305d\u306e\u7d42\u57df\u3092\u5024\u57df\u306b\u307e\u3067\u5236\u9650\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u5168\u5c04\u3092\u8a98\u5c0e\u3057\u3001\u4efb\u610f\u306e\u5168\u5c04\u306f\u540c\u3058\u6c7a\u307e\u3063\u305f\u5024\u306b\u5199\u308b\u3088\u3046\u306a\u5b9a\u7fa9\u57df\u306e\u5143\u3092\u540c\u4e00\u8996\u3057\u3066\u6f70\u3059\u3088\u3046\u306a\u5546\u96c6\u5408\u306e\u4e0a\u306e\u5168\u5358\u5c04\u3092\u8a98\u5c0e\u3059\u308b\u3002\u304d\u3061\u3093\u3068\u8ff0\u3079\u308c\u3070\u3001\u4efb\u610f\u306e\u5168\u5c04 f: A \u2192 B \u306f\u4ee5\u4e0b\u306b\u8ff0\u3079\u308b\u3088\u3046\u306b\u5168\u5358\u5c04\u3068\u5c04\u5f71\u306e\u5408\u6210\u306b\u5206\u89e3\u3055\u308c\u308b\u3002A\/\u223c \u3092 x \u223c y \u21d4 f(x) = f(y) \u3067\u5b9a\u3081\u3089\u308c\u308b\u540c\u5024\u95a2\u4fc2\u306b\u3088\u308b A \u306e\u540c\u5024\u985e\u5168\u4f53\u306e\u6210\u3059\u96c6\u5408\u3068\u3059\u308b\u3002A\/\u223c \u3092 f \u306b\u3088\u308b\u539f\u50cf\u5168\u4f53\u306e\u6210\u3059\u96c6\u5408\u3068\u3059\u308b\u3068\u3044\u3063\u3066\u3082\u540c\u3058\u3053\u3068\u3067\u3042\u308b\u3002\u5199\u50cf P~: A \u21a0 A\/\u223c \u3092 A \u306e\u5404\u5143 x \u3092\u305d\u306e\u540c\u5024\u985e [x]~ \u3078\u5199\u3059\u5c04\u5f71\u3068\u3057\u3001fP: A\/\u223c \u2192 B \u3092  fP([x]\u223c):=f(x){displaystyle f_{P}([x]_{sim }):=f(x)} \u3067\u4e0e\u3048\u3089\u308c\u308b\u3088\u304f\u5b9a\u7fa9\u3055\u308c\u305f\u5199\u50cf\u3068\u3059\u308c\u3070\u3053\u308c\u306f\u5168\u5358\u5c04\u3067\u3001f = fP \u2218 P~ \u304c\u6210\u308a\u7acb\u3064\u3002\u6570\u3048\u4e0a\u3052[\u7de8\u96c6]\u5305\u9664\u539f\u7406\u306e\u5fdc\u7528\u3068\u3057\u3066\u3001\u6709\u9650\u96c6\u5408 X \u304b\u3089 Y \u3078\u306e\u5168\u5c04\u306e\u6570\u306fn!S(m,n)=\u2211k=0n(\u22121)k(nk)(n\u2212k)m{displaystyle n!S(m,n)=sum _{k=0}^{n}(-1)^{k}{binom {n}{k}}(n-k)^{m}}\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b[2]\u3002\u3053\u3053\u3067 m, n \u306f\u6709\u9650\u96c6\u5408 X, Y \u306e\u6fc3\u5ea6\u3067\u3042\u308a\u3001S(m, n) \u306f\u7b2c\u4e8c\u7a2e\u30b9\u30bf\u30fc\u30ea\u30f3\u30b0\u6570\u3067\u3042\u308b\u3002\u6ce8\u91c8[\u7de8\u96c6]^ \u5168\u5c04\u306e\u4ee3\u308f\u308a\u306b\u300c\u4e0a\u3078\u306e\u300d\u3068\u3044\u3046\u8a00\u8449\u3092\u7528\u3044\u308b\u6587\u732e\u3067\u306f\u3001\u5358\u5c04\u306e\u4ee3\u308f\u308a\u306b\u300c\u4e00\u5bfe\u4e00\u300d(one-to-one) \u3068\u3044\u3046\u8a00\u8449\u304c\u4f7f\u308f\u308c\u308b\u304c\u3001\u5f8c\u8005\u306f\u5168\u5358\u5c04\u3092\u8868\u3059\u300c\u4e00\u5bfe\u4e00\u5bfe\u5fdc (one-to-one correspondence)\u300d\u3068\u307e\u304e\u3089\u308f\u3057\u3044\u3002\u5bb9\u6613\u306b\u985e\u63a8\u3055\u308c\u308b\u3088\u3046\u306b\u300c\u4e2d\u3078\u306e\u300d(into) \u3068\u3044\u3046\u8a00\u8449\u304c\u5168\u5c04\u3067\u306a\u3044\u5199\u50cf\u3092\u8868\u3059\u306e\u306b\u7528\u3044\u3089\u308c\u308b\u5834\u5408\u304c\u7a00\u306b\u3042\u308b\uff08\u4f8b\u3048\u3070\u3001\u30b1\u30ea\u30fc (1968),\u5f4c\u6c38 & \u5c0f\u5e73 (1961)\uff09\u3002\u4f53\u306e\u6e96\u540c\u578b\uff08\u3053\u308c\u306f\u5e38\u306b\u5358\u5c04\uff09\u304c\u5168\u5c04\uff08\u5f93\u3063\u3066\u540c\u578b\uff09\u3067\u306a\u3044\u3068\u304d\u3001\u300c\u4e2d\u3078\u306e\u540c\u578b\u300d\u3068\u547c\u3076\u3053\u3068\u306f\u3088\u304f\u3042\u308b\u3002\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/113090#breadcrumbitem","name":"\u5168\u5c04 – Wikipedia"}}]}]