{"id":9253,"date":"2025-10-17T19:57:43","date_gmt":"2025-10-17T10:57:43","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9253"},"modified":"2026-09-27T11:43:49","modified_gmt":"2026-09-27T02:43:49","slug":"ch05-infinite-sets","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\/","title":{"rendered":"\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569"},"content":{"rendered":"
<\/p>\n

<\/p>\n

2\uc7a5\uc5d0\uc11c \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc744 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c \uc9c1\uad00\uc801\uc73c\ub85c \uad6c\ubd84\ud558\uc600\uace0, 4\uc7a5\uc5d0\uc11c\ub294 \ub450 \uc9d1\ud569 \uc0ac\uc774\uc758 \uc77c\ub300\uc77c\ub300\uc751\uacfc \ub300\ub4f1\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \ubb34\ud55c\uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud558\uace0, \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\uc9d1\ud569\uc744 \uad6c\ubcc4\ud55c\ub2e4. \ud2b9\ud788 \uc11c\ub85c \ub2e4\ub978 \ud06c\uae30\uc758 \ubb34\ud55c\uc9d1\ud569\uc774 \uc874\uc7ac\ud568\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

1. \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc758 \uc815\uc758<\/h3>\n

\uc9d1\ud569 \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569<\/span>(finite set)\uc774\ub77c\ub294 \uac83\uc740 \\(A=\\varnothing\\)\uc774\uac70\ub098, \uc5b4\ub5a4 \uc591\uc758 \uc815\uc218 \\(k\\)\uc640 \uc11c\ub85c \ub2e4\ub978 \uc6d0\uc18c \\(a_1,\\,a_2,\\,\\ldots,\\,a_k\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[A=\\{a_1,\\,a_2,\\,\\ldots,\\,a_k\\}\\]
\n\ub85c \uc4f8 \uc218 \uc788\ub2e4\ub294 \ub73b\uc774\ub2e4. \uc774\ub54c \\(k\\)\ub97c \uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218<\/span> \ub610\ub294 \ud06c\uae30<\/span>\ub77c\uace0 \ud558\uace0 \\(|A|=k\\) \ub610\ub294 \\(n(A)=k\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788 \\(|\\varnothing|=0\\)\uc73c\ub85c \ub454\ub2e4.<\/p>\n

\uc9d1\ud569 \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774 \uc544\ub2d0 \ub54c \\(A\\)\ub97c \ubb34\ud55c\uc9d1\ud569<\/span>(infinite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ubb34\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \uc720\ud55c\ud55c \ubaa9\ub85d\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubaa8\ub450 \ub098\uc5f4\ud560 \uc218 \uc5c6\ub2e4. \uadf8\ub7ec\ub098 \ub4a4\uc5d0\uc11c \ubcfc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\ucc98\ub7fc \ubb34\ud55c\ud55c \uc218\uc5f4
\n\\[a_0,\\,a_1,\\,a_2,\\,\\ldots\\]
\n\uc758 \ud615\ud0dc\ub85c \ube60\uc9d0\uc5c6\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub294 \ubb34\ud55c\uc9d1\ud569\ub3c4 \uc788\ub2e4.<\/p>\n

\uc608\ub97c \ub4e4\uc5b4 \uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}=\\{0,\\,1,\\,2,\\,3,\\,\\ldots\\}\\)\uacfc \uadf8 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc778 \uc9dd\uc218 \uc790\uc5f0\uc218\uc758 \uc9d1\ud569 \\(\\{0,\\,2,\\,4,\\,6,\\,\\ldots\\}\\) \uc0ac\uc774\uc5d0\ub294 \\(n\\mapsto 2n\\)\uc774\ub77c\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\ucc98\ub7fc \uc5b4\ub5a4 \ubb34\ud55c\uc9d1\ud569\uc740 \uc790\uae30 \uc9c4\ubd80\ubd84\uc9d1\ud569\uacfc \ub300\ub4f1\ud558\ub2e4. \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc5d0 \ub300\ud558\uc5ec \uac19\uc740 \uc8fc\uc7a5\uc774 \uc131\ub9bd\ud558\ub294\uc9c0\ub294 \uc774 \uc7a5 \ub4a4\uc5d0\uc11c \ub2e4\uc2dc \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

2. \uc77c\ub300\uc77c \ub300\uc751\uacfc \uc9d1\ud569\uc758 \ud06c\uae30<\/h3>\n

4\uc7a5\uc5d0\uc11c \uc815\uc758\ud55c \uac83\ucc98\ub7fc \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c \\(A\\)\uc640 \\(B\\)\uac00 \ub300\ub4f1\ud558\ub2e4\uace0 \ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub97c
\n\\[A\\sim B\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ubb38\uc81c 4.16\uc5d0\uc11c \ubcf4\uc558\ub4ef\uc774 \ub300\ub4f1\uc740 \ubc18\uc0ac\uc801, \ub300\uce6d\uc801, \ucd94\uc774\uc801\uc774\ub2e4.<\/p>\n

\ub450 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud558\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c \ud45c\uae30\ub3c4 \uc0ac\uc6a9\ud55c\ub2e4. \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\uac00 \uc874\uc7ac\ud560 \ub54c
\n\\[A\\preccurlyeq B\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(A\\preccurlyeq B\\)\uc774\uc9c0\ub9cc \\(A\\not\\sim B\\)\uc774\uba74 \\(A\\prec B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud45c\uae30\ub294 7\uc7a5\uc5d0\uc11c \uae30\uc218\ub97c \uc815\uc758\ud55c \ub4a4 \uae30\uc218\uc758 \ub300\uc18c\uad00\uacc4\uc640 \uc5f0\uacb0\ub41c\ub2e4.<\/p>\n

\n

\ubb38\uc81c 5.1.<\/span>
\n\uc9d1\ud569
\n\\[A=\\{1,2,3\\},\\quad B=\\{a,b,c,d\\},\\quad C=\\{0,2,4\\}\\]
\n\ub97c \uc0dd\uac01\ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

    \n
  1. \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218 \ud558\ub098\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
  2. \\(A\\)\uc5d0\uc11c \\(C\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ub300\uc751 \ud558\ub098\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
  3. \\(A\\preccurlyeq B\\), \\(A\\prec B\\), \\(A\\sim C\\) \uc911 \ucc38\uc778 \uac83\uc744 \ubaa8\ub450 \ucc3e\uace0 \uadf8 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
  4. \\(B\\preccurlyeq A\\)\uc778\uc9c0 \ud310\uc815\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

    \ub2e4\uc74c \uacb0\uacfc\ub97c \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac<\/span>(Cantor–Bernstein theorem)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 5.1. (\uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac)<\/span><\/p>\n

    \uc9d1\ud569 \\(A\\), \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\preccurlyeq B\\)\uc774\uace0 \\(B\\preccurlyeq A\\)\uc774\uba74 \\(A\\sim B\\)\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 5.2.<\/span>
    \n\ub2e4\uc74c \uacfc\uc815\uc744 \ud1b5\ud574 \uc815\ub9ac 5.1\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

    \uc77c\ub300\uc77c\ud568\uc218 \\(f\\colon A\\to B\\), \\(g\\colon B\\to A\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0
    \n\\[A_0=A\\setminus g(B),\\quad A_{n+1}=g(f(A_n)),\\quad C=\\bigcup_{n\\in\\mathbb{N}}A_n\\]
    \n\uc73c\ub85c \ub454\ub2e4. \\(a\\in C\\)\uc774\uba74 \\(h(a)=f(a)\\)\ub85c \ub450\uace0, \\(a\\notin C\\)\uc774\uba74 \\(g(b)=a\\)\uc778 \uc720\uc77c\ud55c \\(b\\in B\\)\uc5d0 \ub300\ud558\uc5ec \\(h(a)=b\\)\ub85c \ub450\uc5c8\uc744 \ub54c \\(h\\colon A\\to B\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc784\uc744 \ubcf4\uc778\ub2e4.<\/p>\n<\/div>\n

    3. \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569<\/h3>\n

    \uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud55c \ubb34\ud55c\uc9d1\ud569\uc744 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569<\/span>(countably infinite set) \ub610\ub294 \uac00\ubd80\ubc88\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc720\ud55c\uc9d1\ud569\uacfc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc744 \ud1b5\ud2c0\uc5b4 \uac00\uc0b0\uc9d1\ud569<\/span>(countable set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \\(a_0,\\,a_1,\\,a_2,\\,a_3,\\,\\ldots\\)\uc640 \uac19\uc774 \uc790\uc5f0\uc218\ub85c \ubc88\ud638\ub97c \ub9e4\uaca8 \ube60\uc9d0\uc5c6\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \uc774\uc81c \uc815\uc218 \uc9d1\ud569 \\(\\mathbb{Z}\\)\uc640 \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\uac00 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n

    \uc6b0\uc120 \uc815\uc218 \uc9d1\ud569 \\(\\mathbb{Z}\\)\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ub2e4\uc74c \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.
    \n\\[f\\colon\\mathbb{N}\\to\\mathbb{Z},\\quad
    \nf(n)=
    \n\\begin{cases}
    \n\\dfrac{n}{2} & (n\\text{\uc774 \uc9dd\uc218\uc77c \ub54c}),\\\\[10pt]
    \n-\\dfrac{n+1}{2} & (n\\text{\uc774 \ud640\uc218\uc77c \ub54c}).
    \n\\end{cases}\\]
    \n\uc774 \ud568\uc218\ub294 \\(\\mathbb{N}\\)\uc758 \uc6d0\uc18c\ub97c \\(0,\\,-1,\\,1,\\,-2,\\,2,\\,-3,\\,3,\\,\\ldots\\) \uc21c\uc11c\ub85c \\(\\mathbb{Z}\\)\uc758 \uc6d0\uc18c\uc640 \ub300\uc751\uc2dc\ud0a4\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbb{Z}\\)\ub294 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/p>\n

    \n

    \ubb38\uc81c 5.3.<\/span>
    \n\uc704\uc758 \ud568\uc218 \\(f\\colon\\mathbb{N}\\to\\mathbb{Z}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \\(f(0),f(1),\\ldots,f(7)\\)\uc744 \ucc28\ub840\ub85c \uad6c\ud558\uc2dc\uc624.<\/li>\n
    2. \uac01 \\(z\\in\\mathbb{Z}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(n)=z\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(n\\in\\mathbb{N}\\)\uc744 \\(z\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
    3. \ud568\uc218 \\(g\\colon\\mathbb{Z}\\to\\mathbb{N}\\)\uc744 (2)\uc758 \uc2dd\uacfc \uac19\uc774 \uc815\uc758\ud558\uace0, \\(g\\circ f=\\operatorname{id}_{\\mathbb{N}}\\) \ubc0f \\(f\\circ g=\\operatorname{id}_{\\mathbb{Z}}\\)\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

      \ub2e4\uc74c\uc73c\ub85c \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(0\\)\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub294
      \n\\[\\frac{p}{q}\\quad(p\\in\\mathbb{Z}\\setminus\\{0\\},\\ q\\in\\mathbb{N}\\setminus\\{0\\},\\ \\gcd(|p|,q)=1)\\]
      \n\uc758 \uaf34\ub85c \uc720\uc77c\ud558\uac8c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \\(0=0\/1\\)\uc744 \uba3c\uc800 \ub193\uace0, \ub098\uba38\uc9c0 \uae30\uc57d\ubd84\uc218\ub4e4\uc744 \\(|p|+q\\)\uac00 \uc791\uc740 \uac83\ubd80\ud130 \ub098\uc5f4\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc21c\uc11c\ub97c \uc5bb\ub294\ub2e4.
      \n\\[
      \n\\begin{aligned}
      \n&\\frac{0}{1},\\\\
      \n&\\frac{-1}{1},\\ \\frac{1}{1},\\\\
      \n&\\frac{-1}{2},\\ \\frac{1}{2},\\ \\frac{-2}{1},\\ \\frac{2}{1},\\\\
      \n&\\frac{-1}{3},\\ \\frac{1}{3},\\ \\frac{-3}{1},\\ \\frac{3}{1},\\\\
      \n&\\frac{-1}{4},\\ \\frac{1}{4},\\ \\frac{-2}{3},\\ \\frac{2}{3},\\ \\frac{-3}{2},\\ \\frac{3}{2},\\ \\frac{-4}{1},\\ \\frac{4}{1},\\ \\ldots
      \n\\end{aligned}
      \n\\]
      \n\uac01 \uc591\uc758 \uc815\uc218 \\(m\\)\uc5d0 \ub300\ud558\uc5ec \\(|p|+q=m\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uae30\uc57d\ubd84\uc218\ub294 \uc720\ud55c \uac1c\ubfd0\uc774\uace0, \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub294 \uc815\ud655\ud788 \ud55c \uce35\uc5d0 \ub098\ud0c0\ub09c\ub2e4. \ub530\ub77c\uc11c \uac01 \uce35\uc744 \ucc28\ub840\ub85c \uc774\uc5b4 \ubd99\uc774\uba74 \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \uc790\uc5f0\uc218\ub85c \ubc88\ud638 \ub9e4\uaca8 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbb{Q}\\)\ub294 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      \n

      \ubb38\uc81c 5.4.<\/span>
      \n\uc591\uc758 \uc815\uc218 \\(k\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0
      \n\\[A_k=\\mathbb{N}\\times\\{0,1,\\ldots,k-1\\}\\]
      \n\ub85c \ub450\uc790. \ud568\uc218
      \n\\[\\varphi\\colon A_k\\to\\mathbb{N},\\quad \\varphi(n,r)=kn+r\\]
      \n\uc774 \uc77c\ub300\uc77c\ub300\uc751\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \\(A_k\\)\uac00 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc784\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 5.5.<\/span>
      \n\ub2e4\uc74c \uc9d1\ud569\uc774 \uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

        \n
      1. \ubaa8\ub4e0 \uc9dd\uc218 \uc815\uc218\uc758 \uc9d1\ud569<\/li>\n
      2. \ubaa8\ub4e0 \ud640\uc218 \uc815\uc218\uc758 \uc9d1\ud569<\/li>\n
      3. \\(\\mathbb{N}\\times\\mathbb{N}\\)<\/li>\n
      4. \uc720\ud55c \uac1c\uc758 \uac00\uc0b0\uc9d1\ud569\uc758 \ud569\uc9d1\ud569<\/li>\n<\/ol>\n<\/div>\n
        \n

        \ubb38\uc81c 5.6.<\/span>
        \n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

          \n
        1. \\(A\\subseteq\\mathbb{N}\\)\uc774 \ubb34\ud55c\uc9d1\ud569\uc774\uba74 \\(A\\)\ub294 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n
        2. \\(B\\)\uac00 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774\uba74, \\(B\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(B\\)\uc640 \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n
        3. \\(C\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74, \\(C\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(C\\)\uc640 \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

          4. \ube44\uac00\uc0b0\uc9d1\ud569<\/h3>\n

          \uc790\uc5f0\uc218 \uc9d1\ud569\uacfc \ub300\ub4f1\ud558\uc9c0 \uc54a\uc740 \ubb34\ud55c\uc9d1\ud569\uc744 \ube44\uac00\uc0b0\uc9d1\ud569<\/span>(uncountable set) \ub610\ub294 \ube44\uac00\ubd80\ubc88 \ubb34\ud55c\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

          \uc2e4\uc218 \uc9d1\ud569\uc774 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \uba3c\uc800 \uc5f4\ub9b0\uad6c\uac04 \\((0,1)\\)\uc774 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc790. \uadc0\ub958\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \\((0,1)\\)\uc774 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uba74, \uc774 \uad6c\uac04\uc758 \ubaa8\ub4e0 \uc2e4\uc218\ub97c
          \n\\[r_1,\\,r_2,\\,r_3,\\,r_4,\\,\\ldots\\]
          \n\uc640 \uac19\uc774 \ube60\uc9d0\uc5c6\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

          \uac01 \\(r_i\\)\uc758 \uc2ed\uc9c4\ubc95 \uc18c\uc218 \uc804\uac1c\ub294 \ub05d\uc5d0\uc11c\ubd80\ud130 \\(9\\)\uac00 \ubb34\ud55c\ud788 \ubc18\ubcf5\ub418\uc9c0 \uc54a\ub294 \ud45c\ud604\uc744 \ud0dd\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(0.5\\)\ub294 \\(0.5000\\cdots\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \\(0.4999\\cdots\\)\ub85c \ub098\ud0c0\ub0b4\uc9c0 \uc54a\ub294\ub2e4. \uc774 \uc57d\uc18d \uc544\ub798 \uc18c\uc218 \uc804\uac1c\ub294 \uc720\uc77c\ud558\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f0\uc790.
          \n\\[
          \n\\begin{aligned}
          \nr_1&=0.d_{11}d_{12}d_{13}d_{14}\\cdots,\\\\
          \nr_2&=0.d_{21}d_{22}d_{23}d_{24}\\cdots,\\\\
          \nr_3&=0.d_{31}d_{32}d_{33}d_{34}\\cdots,\\\\
          \nr_4&=0.d_{41}d_{42}d_{43}d_{44}\\cdots,\\\\
          \n&\\ \\vdots
          \n\\end{aligned}
          \n\\]
          \n\uc5ec\uae30\uc11c \\(d_{ij}\\)\ub294 \\(r_i\\)\uc758 \uc18c\uc218\uc810 \uc544\ub798 \\(j\\)\uc9f8 \uc790\ub9ac\uc758 \uc22b\uc790\uc774\ub2e4.<\/p>\n

          \uc774\uc81c
          \n\\[s=0.s_1s_2s_3s_4\\cdots\\]
          \n\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
          \n\\[s_i=
          \n\\begin{cases}
          \n5 & (d_{ii}\\ne 5),\\\\[5pt]
          \n6 & (d_{ii}=5).
          \n\\end{cases}\\]
          \n\uadf8\ub7ec\uba74 \\(s\\in(0,1)\\)\uc774\uace0, \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(s\\)\uc758 \uc18c\uc218\uc810 \uc544\ub798 \\(i\\)\uc9f8 \uc790\ub9ac\uc640 \\(r_i\\)\uc758 \uc18c\uc218\uc810 \uc544\ub798 \\(i\\)\uc9f8 \uc790\ub9ac\uac00 \ub2e4\ub974\ub2e4. \uc120\ud0dd\ud55c \uc18c\uc218 \uc804\uac1c\uc758 \uc720\uc77c\uc131\uc5d0 \ub530\ub77c \\(s\\ne r_i\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(s\\)\ub294 \uc704 \ubaa9\ub85d\uc5d0 \ub4e4\uc5b4 \uc788\uc9c0 \uc54a\uc73c\uba70, \uc774\ub294 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \ub098\uc5f4\ud558\uc600\ub2e4\ub294 \uac00\uc815\uacfc \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\((0,1)\\)\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uc774 \ub17c\uc99d\uc744 \uce78\ud1a0\uc5b4\uc758 \ub300\uac01\uc120 \ub17c\ubc95<\/span>(Cantor’s diagonal argument)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 5.7.<\/span>
          \n\\((0,1)\\)\uc758 \uc6d0\uc18c\ub97c \ub098\uc5f4\ud558\ub824\uace0 \ud588\ub354\ub2c8 \ucc98\uc74c \ub124 \ud56d\uc774 \ub2e4\uc74c\uacfc \uac19\uc558\ub2e4\uace0 \ud558\uc790.
          \n\\[
          \nr_1=0.5142\\cdots,\\quad
          \nr_2=0.2351\\cdots,\\quad
          \nr_3=0.7754\\cdots,\\quad
          \nr_4=0.1032\\cdots.
          \n\\]
          \n\ubcf8\ubb38\uc758 \uaddc\uce59
          \n\\[s_i=5\\quad(d_{ii}\\ne5),\\quad s_i=6\\quad(d_{ii}=5)\\]
          \n\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub300\uac01\uc120 \ub17c\ubc95\uc73c\ub85c \ub9cc\ub4dc\ub294 \uc218 \\(s\\)\uc758 \uc18c\uc218\uc810 \uc544\ub798 \ucc98\uc74c \ub124 \uc790\ub9ac \\(s_1,s_2,s_3,s_4\\)\ub97c \uad6c\ud558\uc2dc\uc624. \ub610\ud55c \ub4a4\uc758 \uc790\ub9bf\uac12\uc774 \uc5b4\ub5bb\uac8c \uc815\ud574\uc9c0\ub4e0 \uc774 \\(s\\)\uac00 \\(r_1,r_2,r_3,r_4\\)\uc640 \uac01\uac01 \ub2e4\ub978 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \ud55c\ud3b8
          \n\\[f\\colon(0,1)\\to\\mathbb{R},\\quad f(x)=\\tan\\left(\\pi x-\\frac{\\pi}{2}\\right)\\]
          \n\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\mathbb{R}\\sim(0,1)\\)\uc774\uace0, \\(\\mathbb{R}\\)\ub3c4 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 5.8.<\/span>
          \n\uc9d1\ud569 \\(A\\)\uac00 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uace0 \uc9d1\ud569 \\(B\\)\uac00 \\(A\\)\uc640 \ub300\ub4f1\ud560 \ub54c, \uc9d1\ud569 \\(B\\)\ub3c4 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc9d1\ud569\ub4e4\uc758 \ud06c\uae30\ub97c \ub354 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

          \n

          \uc815\ub9ac 5.2. (\uc5f0\uc18d\uccb4\uc640 \ub300\ub4f1\ud55c \uc9d1\ud569\ub4e4)<\/span><\/p>\n

          \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[[0,1]\\sim(0,1)\\sim\\mathbb{R}\\sim\\mathbb{R}^n\\]
          \n\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\ud3ec\ud568\ud568\uc218 \\((0,1)\\to[0,1]\\)\uc740 \uc77c\ub300\uc77c\ud568\uc218\uc774\uace0, \\(x\\mapsto (x+1)\/3\\)\uc740 \\([0,1]\\)\uc5d0\uc11c \\((0,1)\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\uc5d0 \uc758\ud574 \\([0,1]\\sim(0,1)\\)\uc774\ub2e4. \uc704\uc758 \ud0c4\uc820\ud2b8 \ud568\uc218\ub85c\ubd80\ud130 \\((0,1)\\sim\\mathbb{R}\\)\ub3c4 \uc5bb\ub294\ub2e4. \ub610\ud55c \ud3ec\ud568\ud568\uc218 \\([0,1)\\to[0,1]\\)\uacfc \uc77c\ub300\uc77c\ud568\uc218 \\(x\\mapsto x\/2\\)\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\uc2dc \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \\([0,1]\\sim[0,1)\\)\uc774\ub2e4.<\/p>\n

          \uc774\uc81c \\([0,1]^2\\sim[0,1]\\)\uc784\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 \\([0,1]\\sim[0,1)\\)\uc774\ubbc0\ub85c \\([0,1]^2\\sim[0,1)^2\\)\uc774\ub2e4. \\(x,y\\in[0,1)\\)\uc758 \uc2ed\uc9c4 \uc804\uac1c\ub97c \ub05d\uc5d0\uc11c\ubd80\ud130 \\(9\\)\uac00 \ubb34\ud55c\ud788 \ubc18\ubcf5\ub418\uc9c0 \uc54a\ub3c4\ub85d
          \n\\[x=0.x_1x_2\\cdots,\\quad y=0.y_1y_2\\cdots\\]
          \n\ub85c \ud0dd\ud558\uace0 \\(c_k=10x_k+y_k\\in\\{0,1,\\ldots,99\\}\\)\ub85c \ub454\ub2e4. \ub2e4\uc74c \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.
          \n\\[G(x,y)=\\sum_{k=1}^{\\infty}\\frac{c_k}{100^k}.\\]
          \n\uc218\uc5f4 \\((c_k)\\)\ub294 \ub05d\uc5d0\uc11c\ubd80\ud130 \\(99\\)\uac00 \ubb34\ud55c\ud788 \ubc18\ubcf5\ub418\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc704 \uc2dd\uc740 \\([0,1)\\)\uc758 \ud45c\uc900\uc801\uc778 \\(100\\)\uc9c4 \uc804\uac1c\ub97c \uc815\ud55c\ub2e4. \ub530\ub77c\uc11c \\(G(x,y)=G(x’,y’)\\)\uc774\uba74 \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(c_k=c’_k\\)\uc774\uace0, \ub530\ub77c\uc11c \\(x_k=x’_k\\), \\(y_k=y’_k\\)\uc774\ub2e4. \uc989 \\(G\\)\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ub2e4. \ubc18\ub300\ub85c \\(t\\mapsto(t,0)\\)\uc740 \\([0,1)\\)\uc5d0\uc11c \\([0,1)^2\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ubbc0\ub85c \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\uc5d0 \uc758\ud574 \\([0,1)^2\\sim[0,1)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\mathbb{R}^2\\sim\\mathbb{R}\\)\uc774\ub2e4.<\/p>\n

          \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(\\mathbb{R}^2\\sim\\mathbb{R}\\)\uc744 \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uba74 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\mathbb{R}^n\\sim\\mathbb{R}\\)\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

          \ud2b9\ud788 \\([0,1]\\), \uc2e4\uc218 \uc9c1\uc120 \\(\\mathbb{R}\\), \ud3c9\uba74 \\(\\mathbb{R}^2\\), \uadf8\ub9ac\uace0 \uc784\uc758\uc758 \uc720\ud55c\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04 \\(\\mathbb{R}^n\\)\uc740 \ubaa8\ub450 \ub300\ub4f1\ud558\ub2e4. 3\ucc28\uc6d0 \uacf5\uac04\uc758 \ubc18\uc9c1\uc120\ub3c4 \ub9e4\uac1c\ubcc0\uc218 \\(t\\in[0,\\infty)\\)\ub97c \uc0ac\uc6a9\ud558\uba74 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud568\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 5.9.<\/span>
          \n\ub3c4\ud615\uc744 \uc810\uc758 \uc9d1\ud569\uc73c\ub85c \uac04\uc8fc\ud558\uc600\uc744 \ub54c, \uc591\uc758 \uae38\uc774\ub97c \uac16\ub294 \ub450 \uc120\ubd84\uc774 \uc11c\ub85c \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \uc591\uc758 \uae38\uc774\ub97c \uac16\ub294 \uc120\ubd84\uacfc \uc2e4\uc218 \uc9c1\uc120\uc774 \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \ubcf5\uc18c\uc218 \uc9d1\ud569 \\(\\mathbb{C}\\)\ub3c4 \uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud558\ub2e4. \uc2e4\uc81c\ub85c
          \n\\[\\mathbb{C}=\\{a+bi\\mid a,b\\in\\mathbb{R}\\}\\]
          \n\uc774\uace0, \\((a,b)\\mapsto a+bi\\)\ub294 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \\(\\mathbb{C}\\)\ub85c \uac00\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \uc815\ub9ac 5.2\uc5d0 \uc758\ud574 \\(\\mathbb{R}^2\\sim\\mathbb{R}\\)\uc774\ubbc0\ub85c \\(\\mathbb{C}\\sim\\mathbb{R}\\)\uc774\ub2e4.<\/p>\n

          5. \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569<\/h3>\n

          \uc9d1\ud569 \\(A\\)\uac00 \uc790\uae30 \uc790\uc2e0\uc758 \uc5b4\ub5a4 \uc9c4\ubd80\ubd84\uc9d1\ud569\uacfc \ub300\ub4f1\ud560 \ub54c \\(A\\)\ub97c \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569<\/span>(Dedekind-infinite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc740 \ubaa8\ub450 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc740 \ubc18\ub4dc\uc2dc \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/p>\n

          \ubc18\ub300\ub85c \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc774 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc774\ub77c\ub294 \uba85\uc81c\ub294 ZF \uacf5\ub9ac\uacc4\ub9cc\uc73c\ub85c\ub294 \uc99d\uba85\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc774 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. [ZF\uc640 ZFC \uacf5\ub9ac\uacc4 \ubc0f \uc120\ud0dd\uacf5\ub9ac\ub294 \ub4a4\uc758 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860 \ubd80\ubd84\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.] \ub530\ub77c\uc11c \uc774 \uc7a5\uc758 \ucc98\uc74c\uc5d0 \ubcf8 \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uc131\uc9c8\uc744 \uc544\ubb34 \uac00\uc815 \uc5c6\uc774 \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc5d0 \uc801\uc6a9\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 5.10.<\/span>
          \n\ub2e4\uc74c\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc744 \uc9c1\uc811 \ud655\uc778\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \uc9d1\ud569\uc774 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

            \n
          1. \\(f\\colon\\mathbb{N}\\to\\mathbb{N}\\setminus\\{0\\}\\), \\(f(n)=n+1\\)<\/li>\n
          2. \\(g\\colon\\mathbb{Z}\\to\\mathbb{Z}\\setminus\\{0\\}\\),
            \n\\[
            \ng(n)=
            \n\\begin{cases}
            \nn & (n<0),\\\\[5pt]\nn+1 & (n\\ge0).\n\\end{cases}\n\\]\n<\/li>\n<\/ol>\n

            \uac01 \uacbd\uc6b0\uc5d0 \uacf5\uc5ed\uc774 \uc6d0\ub798 \uc9d1\ud569\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc784\ub3c4 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

            \n

            \ubb38\uc81c 5.11.<\/span>
            \n\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ub450 \uc870\uac74\uc774 \ub3d9\uce58\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

              \n
            1. \\(A\\)\ub294 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n
            2. \\(A\\)\ub294 \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud55c \ubd80\ubd84\uc9d1\ud569\uc744 \uac16\ub294\ub2e4.<\/li>\n<\/ol>\n

              \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \ubaa8\ub4e0 \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc774 \ub370\ub370\ud0a8\ud2b8 \ubb34\ud55c\uc9d1\ud569\uc784\uc744 \ub2e4\uc2dc \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              6. \uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac<\/h3>\n

              \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ube44\uac00\uc0b0\uc9d1\ud569\uc740 \ubaa8\ub450 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud55c \uac83\uc774\uc5c8\ub2e4. \uadf8\ub807\ub2e4\uba74 \ubaa8\ub4e0 \ube44\uac00\uc0b0\uc9d1\ud569\uc774 \\(\\mathbb{R}\\)\uacfc \ub300\ub4f1\ud560\uae4c? \uadf8\ub807\uc9c0 \uc54a\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 5.3. (\uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac)<\/span><\/p>\n

              \uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[A\\prec\\mathcal{P}(A)\\]
              \n\uc774\ub2e4. \ud2b9\ud788 \\(A\\)\uc640 \uadf8 \uba71\uc9d1\ud569 \\(\\mathcal{P}(A)\\) \uc0ac\uc774\uc5d0\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\ud568\uc218 \\(a\\mapsto\\{a\\}\\)\ub294 \\(A\\)\uc5d0\uc11c \\(\\mathcal{P}(A)\\)\ub85c \uac00\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ubbc0\ub85c \\(A\\preccurlyeq\\mathcal{P}(A)\\)\uc774\ub2e4.<\/p>\n

              \uc774\uc81c \uc784\uc758\uc758 \ud568\uc218 \\(f\\colon A\\to\\mathcal{P}(A)\\)\ub97c \uc0dd\uac01\ud558\uace0
              \n\\[B=\\{x\\in A\\mid x\\notin f(x)\\}\\]
              \n\ub85c \ub454\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\ub77c\uba74 \uc5b4\ub5a4 \\(a\\in A\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f(a)=B\\)\uc774\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(B\\)\uc758 \uc815\uc758\uc5d0\uc11c
              \n\\[a\\in B\\quad\\Longleftrightarrow\\quad a\\notin f(a)=B\\]
              \n\ub97c \uc5bb\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(A\\)\uc5d0\uc11c \\(\\mathcal{P}(A)\\)\ub85c \uac00\ub294 \uc704\ub85c\uc758 \ud568\uc218\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba70, \ud2b9\ud788 \uc77c\ub300\uc77c\ub300\uc751\ub3c4 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A\\prec\\mathcal{P}(A)\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 5.12.<\/span>
              \n\uc9d1\ud569 \\(A=\\{1,2,3,4\\}\\)\uc640 \ud568\uc218 \\(f\\colon A\\to\\mathcal{P}(A)\\)\uac00
              \n\\[
              \nf(1)=\\{1,2\\},\\quad f(2)=\\{1\\},\\quad f(3)=\\{3\\},\\quad f(4)=\\varnothing
              \n\\]
              \n\uc774\ub77c\uace0 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

                \n
              1. \\(B=\\{x\\in A\\mid x\\notin f(x)\\}\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
              2. \uac01 \\(i\\in A\\)\uc5d0 \ub300\ud558\uc5ec \\(B\\)\uc640 \\(f(i)\\)\uac00 \uc6d0\uc18c \\(i\\)\uc758 \uc18c\uc18d \uc5ec\ubd80\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub984\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n
              3. \\(B\\)\uac00 \\(f\\)\uc758 \uce58\uc5ed\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc74c\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

                \uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac\ub97c \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uba74
                \n\\[\\mathbb{N}\\prec\\mathcal{P}(\\mathbb{N})\\prec\\mathcal{P}(\\mathcal{P}(\\mathbb{N}))\\prec\\cdots\\]
                \n\uc744 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \ubb34\ud55c\uc9d1\ud569\uc5d0\ub3c4 \uc11c\ub85c \ub2e4\ub978 \uc5ec\ub7ec \ud06c\uae30\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

                \n

                \ubb38\uc81c 5.13.<\/span>
                \n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

                  \n
                1. \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569 \\(A\\)\uc640 \uc720\ud55c\uc9d1\ud569 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\cup B\\sim A\\)\uc774\ub2e4.<\/li>\n
                2. \uad6c\uac04 \\((0,1)\\)\uacfc \\((0,\\infty)\\)\ub294 \ub300\ub4f1\ud558\ub2e4.<\/li>\n
                3. \ubb34\ub9ac\uc218 \uc9d1\ud569\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n
                4. \uc815\uc218\uacc4\uc218\uc778 \uc601\uc774 \uc544\ub2cc \ub2e4\ud56d\uc2dd\uc758 \uadfc\uc774 \ub418\ub294 \ubcf5\uc18c\uc218\ub97c \ub300\uc218\uc801 \uc218<\/span>(algebraic number)\ub77c\uace0 \ud55c\ub2e4. \ub300\uc218\uc801 \uc218\uc758 \uc9d1\ud569\uc740 \uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                  \n

                  \ubb38\uc81c 5.14.<\/span>
                  \n\uc2e4\uc218 \uc9d1\ud569 \\(\\mathbb{R}\\)\uacfc \uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \uba71\uc9d1\ud569 \\(\\mathcal{P}(\\mathbb{N})\\)\uc774 \ub300\ub4f1\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \ub2e4\uc74c \ub450 \uc77c\ub300\uc77c\ud568\uc218\uc640 \uce78\ud1a0\uc5b4-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\ub97c \uc774\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n