{"id":9250,"date":"2025-10-17T19:43:49","date_gmt":"2025-10-17T10:43:49","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9250"},"modified":"2026-09-27T11:43:45","modified_gmt":"2026-09-27T02:43:45","slug":"ch04-relations-and-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\/","title":{"rendered":"\uad00\uacc4\uc640 \ud568\uc218"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc55e \uc7a5\uc5d0\uc11c\ub294 \uc21c\uc11c\uc30d\uacfc \ub370\uce74\ub974\ud2b8 \uacf1\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uc774\uc81c \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc6d0\uc18c \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \ub098\ud0c0\ub0b4\uace0, \uadf8\uc911 \ud2b9\ubcc4\ud55c \ud615\ud0dc\ub97c \uac16\ub294 \ud568\uc218\uc758 \uac1c\ub150\uc73c\ub85c \ub098\uc544\uac04\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uad00\uacc4\uc640 \ud568\uc218\uc758 \uc815\uc758\uc640 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

1. \uad00\uacc4\uc758 \uc815\uc758<\/h3>\n

\uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc5d0 \ub300\ud558\uc5ec, \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uad00\uacc4<\/span>(relation) \\(R\\)\uc740 \ub370\uce74\ub974\ud2b8 \uacf1 \\(A\\times B\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \uc989,
\n\\[R\\subseteq A\\times B\\]
\n\uc77c \ub54c \\(R\\)\uc744 \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uad00\uacc4\ub77c\uace0 \ubd80\ub978\ub2e4. \\((a,b)\\in R\\)\uc77c \ub54c, \u201c\\(a\\)\uc640 \\(b\\) \uc0ac\uc774\uc5d0 \\(R\\)-\uad00\uacc4\uac00 \uc788\ub2e4\u201d \ub610\ub294 \u201c\\(a\\)\uc640 \\(b\\)\uac00 \uad00\uacc4 \\(R\\)\uc5d0 \uc788\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc744 \\(aRb\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\ud2b9\ud788 \\(A=B\\)\uc778 \uacbd\uc6b0, \\(R\\subseteq A\\times A\\)\ub97c \\(A\\) \uc704\uc758 \uad00\uacc4\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc608\ub97c \ub4e4\uc5b4, \\(A=\\{1,\\,2,\\,3\\}\\)\uc77c \ub54c, \u201c\uc791\uac70\ub098 \uac19\ub2e4\u201d \uad00\uacc4\ub97c \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[R=\\{(1,\\,1),\\,(1,\\,2),\\,(1,\\,3),\\,(2,\\,2),\\,(2,\\,3),\\,(3,\\,3)\\}.\\]<\/p>\n

\n

\ubb38\uc81c 4.1.<\/span>
\n\ub2e4\uc74c \uad00\uacc4\ub97c \uc9d1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624. \uc720\ud55c\ud55c \uad00\uacc4\ub294 \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c, \ubb34\ud55c\ud55c \uad00\uacc4\ub294 \uc870\uac74\uc81c\uc2dc\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/p>\n

    \n
  1. \\(A\\)\uac00 \\(10\\) \uc774\ud558\uc778 \uc591\uc758 \uc815\uc218\uc758 \ubaa8\uc784\uc774\uace0, \\(R\\)\uc774 \\(A\\) \uc704\uc758 \uad00\uacc4\uc774\uba70, \\((n,m)\\in R\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \u201c\\(n\\)\uc774 \\(m\\)\uc758 \uc57d\uc218\uc774\ub2e4\u201d\ub77c\uace0 \uc815\uc758\ud588\uc744 \ub54c, \uad00\uacc4 \\(R\\).<\/li>\n
  2. \\(A\\)\uac00 \\(10\\) \uc774\ud558\uc778 \uc591\uc758 \uc815\uc218\uc758 \ubaa8\uc784\uc774\uace0, \\(E\\)\uac00 \\(A\\) \uc704\uc758 \uad00\uacc4\uc774\uba70, \\((n,m)\\in E\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \u201c\\(n\\)\uacfc \\(m\\)\uc758 \uc591\uc758 \uacf5\uc57d\uc218\uc758 \uac1c\uc218\uac00 \\(2\\)\uc774\ub2e4\u201d\ub77c\uace0 \uc815\uc758\ud588\uc744 \ub54c, \uad00\uacc4 \\(E\\).<\/li>\n
  3. \\(\\mathbb{Z}\\)\uac00 \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc774\uace0, \\(C\\)\uac00 \\(\\mathbb{Z}\\) \uc704\uc758 \uad00\uacc4\uc774\uba70, \\((n,m)\\in C\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \u201c\\(n\\)\uc744 \\(4\\)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uc640 \\(m\\)\uc744 \\(4\\)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uac00 \uac19\ub2e4\u201d\ub77c\uace0 \uc815\uc758\ud588\uc744 \ub54c, \uad00\uacc4 \\(C\\).<\/li>\n
  4. \\(\\mathbb{Z}_+\\)\uac00 \uc591\uc758 \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc774\uace0, \\(R\\)\uc774 \\(\\mathbb{Z}_+\\) \uc704\uc758 \uad00\uacc4\uc774\uba70, \\(x\\)\uc640 \\(y\\)\uac00 \\(y=x^2\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc744 \\(xRy\\)\ub85c \ub098\ud0c0\ub0bc \ub54c, \uad00\uacc4 \\(R\\).<\/li>\n<\/ol>\n<\/div>\n

    \uad00\uacc4 \\(R\\subseteq A\\times B\\)\uc758 \uc815\uc758\uc5ed\uacfc \uce58\uc5ed\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

      \n
    • \uc815\uc758\uc5ed<\/span>(domain): \\(\\operatorname{dom}(R)=\\{a\\in A\\mid \\exists b\\in B,\\ (a,b)\\in R\\}.\\)<\/li>\n
    • \uce58\uc5ed<\/span>(range): \\(\\operatorname{ran}(R)=\\{b\\in B\\mid \\exists a\\in A,\\ (a,b)\\in R\\}.\\)<\/li>\n<\/ul>\n
      \n

      \ubb38\uc81c 4.2.<\/span>
      \n\\(A=\\{1,2,3,4\\}\\), \\(B=\\{a,b,c\\}\\)\uc774\uace0
      \n\\[R=\\{(1,a),(3,b),(4,b)\\}\\subseteq A\\times B\\]
      \n\ub77c\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \\(1Ra\\), \\(2Ra\\), \\(3Rb\\), \\(4Rc\\)\uc758 \ucc38\uacfc \uac70\uc9d3\uc744 \uac01\uac01 \ud310\ubcc4\ud558\uc2dc\uc624.<\/li>\n
      2. \\(\\operatorname{dom}(R)\\)\uacfc \\(\\operatorname{ran}(R)\\)\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n
      3. \\(R\\)\uc774 \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uad00\uacc4\uc774\uc9c0\ub9cc \\(\\operatorname{dom}(R)=A\\)\uc77c \ud544\uc694\ub3c4, \\(\\operatorname{ran}(R)=B\\)\uc77c \ud544\uc694\ub3c4 \uc5c6\ub294 \uc774\uc720\ub97c \uc774 \uc608\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

        2. \ud569\uc131\uad00\uacc4\uc640 \uc5ed\uad00\uacc4<\/h3>\n

        \uad00\uacc4 \\(R\\subseteq A\\times B\\)\uc640 \\(S\\subseteq B\\times C\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ud569\uc131\uad00\uacc4<\/span>(composite relation) \\(S\\circ R\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[S\\circ R=\\{(a,\\,c)\\in A\\times C\\mid \\exists b\\in B,\\ (a,\\,b)\\in R\\text{\uc774\uace0 }(b,\\,c)\\in S\\}.\\]<\/p>\n

        \uad00\uacc4 \\(R\\subseteq A\\times B\\)\uc758 \uc5ed\uad00\uacc4<\/span>(inverse relation) \\(R^{-1}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[R^{-1}=\\{(b,\\,a)\\in B\\times A\\mid (a,\\,b)\\in R\\}.\\]<\/p>\n

        \n

        \ubb38\uc81c 4.3.<\/span>
        \n\\(A=\\{1,\\,2,\\,3\\}\\), \\(B=\\{a,\\,b,\\,c\\}\\), \\(C=\\{u,\\,v\\}\\)\uc774\uace0
        \n\\[
        \nR=\\{(1,\\,a),\\,(1,\\,b),\\,(2,\\,c),\\,(3,\\,b)\\},\\quad
        \nS=\\{(a,\\,u),\\,(b,\\,v),\\,(c,\\,u)\\}
        \n\\]
        \n\ub77c\uace0 \ud558\uc790.<\/p>\n

          \n
        1. \ud569\uc131\uad00\uacc4 \\(S\\circ R\\)\uc744 \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
        2. \uc5ed\uad00\uacc4 \\(R^{-1}\\)\uc640 \\(S^{-1}\\)\ub97c \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
        3. \\((S\\circ R)^{-1}\\)\uc640 \\(R^{-1}\\circ S^{-1}\\)\ub97c \uac01\uac01 \uad6c\ud558\uc5ec \ub450 \uad00\uacc4\uac00 \uac19\uc74c\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 4.4.<\/span>
          \n\\(R\\subseteq A\\times B\\), \\(S\\subseteq B\\times C\\)\uc77c \ub54c \\(S\\circ R\\subseteq A\\times C\\)\uc774\uace0 \\(R^{-1}\\subseteq B\\times A\\)\uc784\uc744 \uc815\uc758\uc5d0\uc11c \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \uc815\ub9ac 4.1. (\uad00\uacc4\uc758 \ud569\uc131\uacfc \uc5ed\uad00\uacc4\uc758 \ubc95\uce59)<\/span><\/p>\n

          \\(R\\subseteq A\\times B\\), \\(S\\subseteq B\\times C\\), \\(T\\subseteq C\\times D\\)\ub77c\uace0 \ud558\uc790. \uad00\uacc4\uc758 \ud569\uc131\uc5d0 \ub300\ud558\uc5ec \uacb0\ud569\ubc95\uce59
          \n\\[(T\\circ S)\\circ R=T\\circ(S\\circ R)\\]
          \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

            \n
          1. \\((R^{-1})^{-1}=R\\)<\/li>\n
          2. \\((S\\circ R)^{-1}=R^{-1}\\circ S^{-1}\\)<\/li>\n
          3. \\(\\operatorname{dom}(R^{-1})=\\operatorname{ran}(R)\\)<\/li>\n
          4. \\(\\operatorname{ran}(R^{-1})=\\operatorname{dom}(R)\\)<\/li>\n<\/ol>\n<\/div>\n
            \n

            \ubb38\uc81c 4.5.<\/span>
            \n\uc815\ub9ac 4.1\uc758 \uc5ed\uad00\uacc4\uc5d0 \uad00\ud55c \ub124 \ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

            \n

            \ubb38\uc81c 4.6.<\/span>
            \n\uad00\uacc4\uc758 \ud569\uc131\uc758 \uacb0\ud569\ubc95\uce59 \\((T\\circ S)\\circ R=T\\circ(S\\circ R)\\)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

            3. \uad00\uacc4\uc758 \uc885\ub958<\/h3>\n

            \\(R\\)\uc774 \uc9d1\ud569 \\(A\\) \uc704\uc758 \uad00\uacc4\ub77c\uace0 \ud558\uc790. \uc989 \\(R\\subseteq A\\times A\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

              \n
            • \\(R\\)\uc774 \ubc18\uc0ac\uc801<\/span>(reflexive) \uad00\uacc4\ub77c \ud568\uc740, \ubaa8\ub4e0 \\(a\\in A\\)\uc5d0 \ub300\ud574 \\((a,a)\\in R\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n
            • \\(R\\)\uc774 \ub300\uce6d\uc801<\/span>(symmetric) \uad00\uacc4\ub77c \ud568\uc740, \\((a,b)\\in R\\)\uc77c \ub54c\ub9c8\ub2e4 \\((b,a)\\in R\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n
            • \\(R\\)\uc774 \ubc18\ub300\uce6d\uc801<\/span>(antisymmetric) \uad00\uacc4\ub77c \ud568\uc740, \\((a,b)\\in R\\)\uc774\uace0 \\((b,a)\\in R\\)\uc77c \ub54c\ub9c8\ub2e4 \\(a=b\\)\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n
            • \\(R\\)\uc774 \ucd94\uc774\uc801<\/span>(transitive) \uad00\uacc4\ub77c \ud568\uc740, \\((a,b)\\in R\\)\uc774\uace0 \\((b,c)\\in R\\)\uc77c \ub54c\ub9c8\ub2e4 \\((a,c)\\in R\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n<\/ul>\n

              \ud2b9\ud788 \uc218\ud559\uc5d0\uc11c \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud558\ub294 \uad00\uacc4\ub294 \ub3d9\uce58\uad00\uacc4\uc640 \uc21c\uc11c\uad00\uacc4\uc774\ub2e4.<\/p>\n

                \n
              • \ubc18\uc0ac\uc801\uc774\uace0 \ub300\uce6d\uc801\uc774\uba70 \ucd94\uc774\uc801\uc778 \uad00\uacc4\ub97c \ub3d9\uce58\uad00\uacc4<\/span>(equivalence relation)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n
              • \ubc18\uc0ac\uc801\uc774\uace0 \ubc18\ub300\uce6d\uc801\uc774\uba70 \ucd94\uc774\uc801\uc778 \uad00\uacc4\ub97c \uc21c\uc11c\uad00\uacc4<\/span>(order relation)\ub77c\uace0 \ubd80\ub978\ub2e4. [\uc5ec\uae30\uc11c \uc21c\uc11c\uad00\uacc4\ub294 \ud754\ud788 \ubd80\ubd84\uc21c\uc11c\uad00\uacc4(partial order)\ub77c\uace0 \ubd80\ub974\ub294 \uac1c\ub150\uc774\ub2e4. \uc784\uc758\uc758 \ub450 \uc6d0\uc18c\uac00 \ube44\uad50 \uac00\ub2a5\ud558\ub2e4\ub294 \uc870\uac74\uae4c\uc9c0 \ub9cc\uc871\uc2dc\ud0a4\uba74 \uc804\uc21c\uc11c(total order)\ub77c\uace0 \ubd80\ub978\ub2e4.]<\/li>\n<\/ul>\n
                \n

                \ubb38\uc81c 4.7.<\/span>
                \n\\(A=\\{1,\\,2,\\,3\\}\\) \uc704\uc758 \ub2e4\uc74c \uc138 \uad00\uacc4\ub97c \uc0dd\uac01\ud558\uc790.
                \n\\[
                \n\\begin{aligned}
                \nR&=\\{(1,\\,1),\\,(2,\\,2),\\,(3,\\,3),\\,(1,\\,2),\\,(2,\\,1)\\},\\\\[3pt]
                \nS&=\\{(1,\\,1),\\,(2,\\,2),\\,(3,\\,3),\\,(1,\\,2),\\,(1,\\,3),\\,(2,\\,3)\\},\\\\[3pt]
                \nT&=\\{(1,\\,1),\\,(2,\\,2),\\,(3,\\,3),\\,(1,\\,2),\\,(2,\\,1),\\,(2,\\,3),\\,(3,\\,2)\\}.
                \n\\end{aligned}
                \n\\]
                \n\uac01 \uad00\uacc4\uac00 \ubc18\uc0ac\uc801, \ub300\uce6d\uc801, \ubc18\ub300\uce6d\uc801, \ucd94\uc774\uc801\uc778\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624. \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294 \uc131\uc9c8\uc5d0 \ub300\ud574\uc11c\ub294 \uad6c\uccb4\uc801\uc778 \ubc18\ub840\ub97c \ud558\ub098\uc529 \uc81c\uc2dc\ud558\uc2dc\uc624. \ub610\ud55c \uac01 \uad00\uacc4\uac00 \ub3d9\uce58\uad00\uacc4\uc778\uc9c0 \ub610\ub294 \uc21c\uc11c\uad00\uacc4\uc778\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                4. \uad00\uacc4\uc758 \uc608<\/h3>\n

                \uc218\ud559\uc5d0\uc11c \uc790\uc8fc \ub4f1\uc7a5\ud558\ub294 \uad00\uacc4\uc758 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

                  \n
                • \uc2e4\uc218 \uc9d1\ud569\uc5d0\uc11c\uc758 \ub4f1\ud638 \uad00\uacc4: \\(R=\\{(x,x)\\mid x\\in\\mathbb{R}\\}\\)\ub294 \ub3d9\uce58\uad00\uacc4\uc774\ub2e4.<\/li>\n
                • \uc2e4\uc218 \uc9d1\ud569\uc5d0\uc11c\uc758 \ubd80\ub4f1\ud638 \uad00\uacc4: \\(R=\\{(x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\mid x\\leq y\\}\\)\ub294 \uc21c\uc11c\uad00\uacc4\uc774\ub2e4.<\/li>\n
                • \uc9d1\ud569\uc871\uc5d0\uc11c\uc758 \ubd80\ubd84\uc9d1\ud569 \uad00\uacc4: \uc9d1\ud569\uc871 \\(\\mathcal{F}\\)\uc5d0 \ub300\ud558\uc5ec \\(R=\\{(A,B)\\in\\mathcal{F}\\times\\mathcal{F}\\mid A\\subseteq B\\}\\)\ub294 \uc21c\uc11c\uad00\uacc4\uc774\ub2e4.<\/li>\n
                • \uc815\uc218 \uc9d1\ud569\uc5d0\uc11c\uc758 \ud569\ub3d9 \uad00\uacc4: \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(R=\\{(a,b)\\in\\mathbb{Z}\\times\\mathbb{Z}\\mid a\\equiv b\\pmod{n}\\}\\)\uc740 \ub3d9\uce58\uad00\uacc4\uc774\ub2e4.<\/li>\n
                • \uc591\uc758 \uc815\uc218 \uc9d1\ud569\uc5d0\uc11c\uc758 \uc57d\uc218 \uad00\uacc4: \\(R=\\{(a,b)\\in\\mathbb{Z}_+\\times\\mathbb{Z}_+\\mid a\\text{\ub294 }b\\text{\uc758 \uc57d\uc218}\\}\\)\ub294 \uc21c\uc11c\uad00\uacc4\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\mathbb{Z}_+\\)\ub294 \uc591\uc758 \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ul>\n
                  \n

                  \ubb38\uc81c 4.8.<\/span>
                  \n\uc704 \uc608\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \uad00\uacc4\uac00 \uac01\uac01 \ub3d9\uce58\uad00\uacc4 \ub610\ub294 \uc21c\uc11c\uad00\uacc4\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                  5. \ub3d9\uce58\ub958\uc640 \ubd84\ud560<\/h3>\n

                  \\(R\\)\uc774 \uc9d1\ud569 \\(A\\) \uc704\uc758 \ub3d9\uce58\uad00\uacc4\uc774\uace0 \\(a\\in A\\)\ub77c\uace0 \ud558\uc790. \\(a\\)\uc758 \ub3d9\uce58\ub958<\/span>(equivalence class)\ub97c
                  \n\\[[a]_R=\\{x\\in A\\mid xRa\\}\\]
                  \n\ub85c \uc815\uc758\ud55c\ub2e4. \ubb38\ub9e5\uc5d0\uc11c \uad00\uacc4 \\(R\\)\uc774 \ubd84\uba85\ud558\uba74 \\([a]\\)\ub77c\uace0 \uac04\ub2e8\ud788 \uc4f0\uae30\ub3c4 \ud55c\ub2e4. \ubaa8\ub4e0 \ub3d9\uce58\ub958\uc758 \uc9d1\ud569
                  \n\\[A\/R=\\{[a]_R\\mid a\\in A\\}\\]
                  \n\uc744 \\(R\\)\uc5d0 \uc758\ud55c \ubaab\uc9d1\ud569<\/span>(quotient set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

                  \uc9d1\ud569 \\(A\\)\uc758 \ubd84\ud560<\/span>(partition)\uc740 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(A\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \uc9d1\ud569\uc871 \\(\\mathcal{P}\\)\uc774\ub2e4.<\/p>\n

                    \n
                  1. \ubaa8\ub4e0 \\(P\\in\\mathcal{P}\\)\uc5d0 \ub300\ud558\uc5ec \\(P\\ne\\varnothing\\)\uc774\uace0, \\(\\displaystyle\\bigcup_{P\\in\\mathcal{P}}P=A\\)\uc774\ub2e4.<\/li>\n
                  2. \uc11c\ub85c \ub2e4\ub978 \\(P,Q\\in\\mathcal{P}\\)\uc5d0 \ub300\ud558\uc5ec \\(P\\cap Q=\\varnothing\\)\uc774\ub2e4.<\/li>\n<\/ol>\n
                    \n

                    \uc815\ub9ac 4.2. (\ub3d9\uce58\uad00\uacc4\uc640 \ubd84\ud560)<\/span><\/p>\n

                    \uc9d1\ud569 \\(A\\) \uc704\uc758 \ub3d9\uce58\uad00\uacc4 \\(R\\)\uc758 \ub3d9\uce58\ub958\ub4e4\uc740 \\(A\\)\uc758 \ubd84\ud560\uc744 \uc774\ub8ec\ub2e4. \ubc18\ub300\ub85c \\(A\\)\uc758 \ubd84\ud560 \\(\\mathcal{P}\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c,
                    \n\\[xRy\\quad\\Longleftrightarrow\\quad x\\text{\uc640 }y\\text{\uac00 }\\mathcal{P}\\text{\uc758 \uac19\uc740 \uc6d0\uc18c\uc5d0 \uc18d\ud55c\ub2e4}\\]
                    \n\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(R\\)\uc740 \\(A\\) \uc704\uc758 \ub3d9\uce58\uad00\uacc4\uc774\uace0, \uadf8 \ub3d9\uce58\ub958\ub4e4\uc740 \uc815\ud655\ud788 \\(\\mathcal{P}\\)\uc758 \uc6d0\uc18c\ub4e4\uc774\ub2e4.<\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 4.9.<\/span>
                    \n\\(A=\\{1,\\,2,\\,3,\\,4,\\,5,\\,6,\\,7,\\,8\\}\\)\uc774\ub77c\uace0 \ud558\uace0, \\(x,y\\in A\\)\uc5d0 \ub300\ud558\uc5ec
                    \n\\[xRy\\quad\\Longleftrightarrow\\quad x\\text{\ub97c }3\\text{\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uc640 }y\\text{\ub97c }3\\text{\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uac00 \uac19\ub2e4}\\]
                    \n\ub77c\uace0 \uc815\uc758\ud558\uc790.<\/p>\n

                      \n
                    1. \\([1]_R\\), \\([2]_R\\), \\([3]_R\\), \\([4]_R\\)\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n
                    2. \ubaab\uc9d1\ud569 \\(A\/R\\)\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n
                    3. \\(A\/R\\)\uc758 \uc6d0\uc18c\ub4e4\uc774 \\(A\\)\uc758 \ubd84\ud560\uc744 \uc774\ub8f8\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                      \n

                      \ubb38\uc81c 4.10.<\/span>
                      \n\uc815\ub9ac 4.2\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                      6. \uc81c\ud55c\ub41c \uad00\uacc4<\/h3>\n

                      \uad00\uacc4 \\(R\\subseteq A\\times A\\)\uc640 \ubd80\ubd84\uc9d1\ud569 \\(B\\subseteq A\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(R\\)\uc744 \\(B\\) \uc704\ub85c \uc81c\ud55c\ud558\uc5ec \uc5bb\uc740 \uc81c\ud55c\ub41c \uad00\uacc4<\/span>(restriction) \\(R|_B\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
                      \n\\[R|_B=R\\cap(B\\times B)=\\{(x,y)\\in R\\mid x,y\\in B\\}.\\]<\/p>\n

                      \uc608\ub97c \ub4e4\uc5b4, \uc2e4\uc218\uc5d0\uc11c\uc758 \uc21c\uc11c\uad00\uacc4 \\(\\leq\\)\ub97c \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\ub85c \uc81c\ud55c\ud558\uba74 \uc720\ub9ac\uc218\uc5d0\uc11c\uc758 \uc21c\uc11c\uad00\uacc4\ub97c \uc5bb\ub294\ub2e4. \uc81c\ud55c\ub41c \uad00\uacc4\ub294 \uc6d0\ub798 \uad00\uacc4\uc758 \uc131\uc9c8\uc744 \uc774\uc5b4\ubc1b\ub294\ub2e4. \uc989, \\(R\\)\uc774 \ubc18\uc0ac\uc801, \ub300\uce6d\uc801, \ubc18\ub300\uce6d\uc801, \ucd94\uc774\uc801 \uad00\uacc4\uc774\uba74 \\(R|_B\\)\ub3c4 \uac01\uac01 \uac19\uc740 \uc131\uc9c8\uc744 \uac16\ub294\ub2e4. \ub530\ub77c\uc11c \ub3d9\uce58\uad00\uacc4\uc758 \uc81c\ud55c\uc740 \ub3d9\uce58\uad00\uacc4\uc774\uace0 \uc21c\uc11c\uad00\uacc4\uc758 \uc81c\ud55c\uc740 \uc21c\uc11c\uad00\uacc4\uc774\ub2e4.<\/p>\n

                      \n

                      \ubb38\uc81c 4.11.<\/span>
                      \n\\(A=\\{1,\\,2,\\,3,\\,4,\\,6,\\,12\\}\\) \uc704\uc5d0\uc11c \\(xRy\\)\ub97c \u201c\\(x\\)\uac00 \\(y\\)\uc758 \uc57d\uc218\uc774\ub2e4\u201d\ub85c \uc815\uc758\ud558\uace0, \\(B=\\{1,\\,2,\\,4,\\,12\\}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

                        \n
                      1. \uc81c\ud55c\ub41c \uad00\uacc4 \\(R|_B\\)\ub97c \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
                      2. \\(R|_B\\)\uac00 \\(B\\) \uc704\uc758 \uc21c\uc11c\uad00\uacc4\uc784\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n
                      3. \\(\\operatorname{dom}(R|_B)\\)\uacfc \\(\\operatorname{ran}(R|_B)\\)\uc744 \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

                        7. \ud568\uc218\uc758 \uc815\uc758<\/h3>\n

                        \uc774 \ucc45\uc5d0\uc11c\ub294 \ud568\uc218<\/span>(function) \\(f\\colon A\\to B\\)\ub97c \uc815\uc758\uc5ed \\(A\\), \uacf5\uc5ed \\(B\\), \uadf8\ub9ac\uace0 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uad00\uacc4 \\(f\\subseteq A\\times B\\)\ub97c \ud568\uaed8 \uc9c0\uc815\ud55c \uac83\uc73c\ub85c \uc774\ud574\ud55c\ub2e4. [\uc774\ub807\uac8c \uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\ub3c4 \ud568\uc218\uc758 \uc790\ub8cc\uc5d0 \ud3ec\ud568\uc2dc\ud0a4\uba74, \uac19\uc740 \ub300\uc751 \uaddc\uce59\uc744 \uc0ac\uc6a9\ud558\ub354\ub77c\ub3c4 \uc815\uc758\uc5ed\uc774\ub098 \uacf5\uc5ed\uc774 \ub2e4\ub974\uba74 \uc11c\ub85c \ub2e4\ub978 \ud568\uc218\uac00 \ub41c\ub2e4.]<\/p>\n

                          \n
                        1. \ud568\uc22b\uac12\uc758 \uc874\uc7ac\uc131: \ubaa8\ub4e0 \\(a\\in A\\)\uc5d0 \ub300\ud558\uc5ec \\((a,b)\\in f\\)\uc778 \\(b\\in B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n
                        2. \ud568\uc22b\uac12\uc758 \uc720\uc77c\uc131: \\((a,b)\\in f\\)\uc774\uace0 \\((a,c)\\in f\\)\uc774\uba74 \\(b=c\\)\uc774\ub2e4.<\/li>\n<\/ol>\n

                          \ub2e4\uc2dc \ub9d0\ud574, \ud568\uc218\ub294 \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \uacf5\uc5ed\uc758 \uc6d0\uc18c\ub97c \uc815\ud655\ud788 \ud558\ub098\uc529 \ub300\uc751\uc2dc\ud0a4\ub294 \uad00\uacc4\uc774\ub2e4.<\/p>\n

                          \\((a,b)\\in f\\)\uc77c \ub54c, \\(b\\)\ub97c \\(a\\)\uc758 \uc0c1<\/span>(image) \ub610\ub294 \\(f\\)\uc5d0 \uc758\ud55c \\(a\\)\uc758 \ud568\uc22b\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(f(a)=b\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

                          \ud568\uc218 \\(f\\colon A\\to B\\)\uc5d0\uc11c \\(A\\)\ub97c \\(f\\)\uc758 \uc815\uc758\uc5ed(domain), \\(B\\)\ub97c \\(f\\)\uc758 \uacf5\uc5ed<\/span>(codomain)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c
                          \n\\[\\{f(x)\\mid x\\in A\\}\\subseteq B\\]
                          \n\ub97c \\(f\\)\uc758 \uce58\uc5ed(range)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud568\uc218\ub85c\uc11c\uc758 \uc815\uc758\uc5ed\uc740 \uad00\uacc4\ub85c \ubcf4\uc558\uc744 \ub54c\uc758 \\(\\operatorname{dom}(f)\\)\uc640 \uac19\uace0, \uce58\uc5ed\uc740 \\(\\operatorname{ran}(f)\\)\uc640 \uac19\ub2e4.<\/p>\n

                          \n

                          \ubb38\uc81c 4.12.<\/span>
                          \n\\(A=\\{1,\\,2,\\,3\\}\\), \\(B=\\{a,\\,b,\\,c\\}\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uac01 \uad00\uacc4\uac00 \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \ud568\uc218\uc778\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624. \ud568\uc218\uc774\uba74 \uac01 \ud568\uc22b\uac12\uacfc \uce58\uc5ed\uc744 \uad6c\ud558\uace0, \ud568\uc218\uac00 \uc544\ub2c8\uba74 \uc874\uc7ac\uc131 \ub610\ub294 \uc720\uc77c\uc131 \uc911 \uc5b4\ub290 \uc870\uac74\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n

                            \n
                          1. \\(R_1=\\{(1,\\,a),\\,(2,\\,b),\\,(3,\\,c)\\}\\)<\/li>\n
                          2. \\(R_2=\\{(1,\\,a),\\,(1,\\,b),\\,(2,\\,c),\\,(3,\\,a)\\}\\)<\/li>\n
                          3. \\(R_3=\\{(1,\\,a),\\,(2,\\,b)\\}\\)<\/li>\n
                          4. \\(R_4=\\{(1,\\,a),\\,(2,\\,a),\\,(3,\\,a)\\}\\)<\/li>\n<\/ol>\n<\/div>\n
                            \n

                            \ubb38\uc81c 4.13.<\/span>
                            \n\uc9d1\ud569 \\(A\\), \\(B\\)\uc640 \uad00\uacc4 \\(R\\)\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c, \\(R\\)\uc774 \\(A\\)\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \ud568\uc218\uac00 \ub418\ub294\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n

                              \n
                            1. \\(A\\)\uc640 \\(B\\)\uac00 \uc2e4\uc218 \uc804\uccb4 \uc9d1\ud569\uc774\uace0, \\((x,y)\\in R\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x=y^2\\)\uc774\ub2e4.<\/li>\n
                            2. \\(A\\)\uc640 \\(B\\)\uac00 \uc591\uc758 \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc774\uace0, \\((x,y)\\in R\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x=y^2\\)\uc774\ub2e4.<\/li>\n
                            3. \\(A\\)\uc640 \\(B\\)\uac00 \uc591\uc758 \uc815\uc218 \uc804\uccb4 \uc9d1\ud569\uc774\uace0, \\((p,q)\\in R\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(p\\)\uac00 \\(q\\)\uc758 \uc57d\uc218\uc778 \uac83\uc774\ub2e4.<\/li>\n
                            4. \\(A\\)\uc640 \\(B\\)\uac00 \uc591\uc758 \uc815\uc218 \uc804\uccb4 \uc9d1\ud569\uc774\uace0, \\((p,q)\\in R\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(12\\)\uc640 \\(p\\)\uc758 \ucd5c\uc18c\uacf5\ubc30\uc218\uac00 \\(q\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

                              8. \ud568\uc218\uc758 \uc885\ub958<\/h3>\n

                              \ud568\uc218 \\(f\\colon A\\to B\\)\ub97c \uc6d0\uc18c\uc758 \ub300\uc751 \uc591\uc0c1\uc5d0 \ub530\ub77c \ub2e4\uc74c\uacfc \uac19\uc774 \ubd84\ub958\ud55c\ub2e4.<\/p>\n