{"id":6978,"date":"2021-07-23T23:19:51","date_gmt":"2021-07-23T14:19:51","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6978"},"modified":"2026-09-27T16:47:35","modified_gmt":"2026-09-27T07:47:35","slug":"infinite-sets","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/infinite-sets\/","title":{"rendered":"\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 3\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\uc74c\uc774 \uc544\ub2cc \uc815\uc218 \\(0,1,2,\\ldots\\)\ub97c \uc9d1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \ud55c \uac00\uc9c0 \ud45c\uc900\uc801\uc778 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[
\n\\begin{align}
\n0 &= \\varnothing,\\\\[6pt]
\n1 &= 0 \\cup \\left\\{ 0 \\right\\} = \\left\\{ 0 \\right\\},\\\\[6pt]
\n2 &= 1 \\cup \\left\\{ 1 \\right\\} = \\left\\{ 0,\\,1 \\right\\},\\\\[6pt]
\n3 &= 2 \\cup \\left\\{ 2 \\right\\} = \\left\\{ 0,\\,1,\\,2 \\right\\},\\\\[6pt]
\n4 &= 3 \\cup \\left\\{ 3 \\right\\} = \\left\\{ 0,\\,1,\\,2,\\,3 \\right\\},\\\\[6pt]
\n&\\qquad\\vdots
\n\\end{align}
\n\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(n\\)\uc774 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\ub77c\uba74, \uc774\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8 \uc9d1\ud569 \\(n\\)\uc740 \\(n\\)\uac1c\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc74c\uc774 \uc544\ub2cc \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744
\n\\[
\n\\mathbb{N}=\\left\\{0,\\,1,\\,2,\\,3,\\,\\ldots\\right\\}
\n\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A\\)\uc640 \\(B\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c \ub300\uc751\uc774 \uc874\uc7ac\ud558\uba74 \u201c\\(A\\)\uc640 \\(B\\)\uc758 \uae30\uc218\uac00 \uac19\ub2e4<\/span>(have the same cardinality)\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c
\n\\[
\n\\lvert A\\rvert=\\lvert B\\rvert
\n\\quad\\text{\ub610\ub294}\\quad
\nA\\approx B
\n\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

<\/p>\n

\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/h2>\n

\\(E\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc74c\uc774 \uc544\ub2cc \uc815\uc218 \\(k\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[
\nE\\approx k
\n\\]
\n\uc774\uba74 \\(E\\)\ub97c \uc720\ud55c\uc9d1\ud569<\/span>(finite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc11c\ub85c \ub2e4\ub978 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\ub294 \uac19\uc740 \uae30\uc218\ub97c \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc774\ub7ec\ud55c \\(k\\)\ub294 \uc720\uc77c\ud558\ub2e4. \uc774\ub54c \\(E\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub294 \\(k\\)\ub77c\uace0 \ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c
\n\\[
\nn(E)=k
\n\\quad\\text{\ub610\ub294}\\quad
\n\\lvert E\\rvert=k
\n\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\uc5b4\ub5a0\ud55c \uc74c\uc774 \uc544\ub2cc \uc815\uc218 \\(k\\)\uc5d0 \ub300\ud574\uc11c\ub3c4 \\(E\\approx k\\)\uac00 \uc131\ub9bd\ud558\uc9c0 \uc54a\uc73c\uba74 \\(E\\)\ub97c \ubb34\ud55c\uc9d1\ud569<\/span>(infinite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uc77c\ub300\uc77c \ud568\uc218\uac00 \uc874\uc7ac\ud558\uba74 \uc774\uac83\uc744 \uae30\ud638\ub85c
\n\\[
\n\\lvert A\\rvert\\le\\lvert B\\rvert
\n\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(\\lvert A\\rvert\\le\\lvert B\\rvert\\)\uc774\uba74\uc11c \\(\\lvert A\\rvert\\ne\\lvert B\\rvert\\)\uc774\uba74
\n\\[
\n\\lvert A\\rvert<\\lvert B\\rvert\n\\]\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub85c\uc368 \uc9d1\ud569\uc758 \uc6d0\uc18c\uac00 \ubb34\ud55c\ud788 \ub9ce\uc744 \ub54c\uc5d0\ub3c4 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\ub2e4\uc74c \uc815\ub9ac\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 0.3.1. (Schr\u00f6der-Bernstein \uc815\ub9ac)<\/span><\/p>\n

\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
\n\\[
\n\\lvert A\\rvert\\le\\lvert B\\rvert
\n\\quad\\text{\uc774\uace0}\\quad
\n\\lvert B\\rvert\\le\\lvert A\\rvert
\n\\]
\n\uc774\uba74
\n\\[
\n\\lvert A\\rvert=\\lvert B\\rvert
\n\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(f:A\\rightarrow B\\)\uc640 \\(g:B\\rightarrow A\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \\(A\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc744 \uc815\uc758\ud55c\ub2e4.
\n\\[
\nA_0=A\\setminus g(B),
\n\\qquad
\nA_{n+1}=g(f(A_n))
\n\\quad(n=0,1,2,\\ldots).
\n\\]
\n\uadf8\ub9ac\uace0
\n\\[
\nC=\\bigcup_{n=0}^{\\infty}A_n
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

\\(a\\in A\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(h:A\\rightarrow B\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[
\nh(a)=
\n\\begin{cases}
\nf(a), & a\\in C,\\\\
\ng^{-1}(a), & a\\notin C.
\n\\end{cases}
\n\\]
\n\uc5ec\uae30\uc11c \\(a\\notin C\\)\uc774\uba74 \ud2b9\ud788 \\(a\\notin A_0=A\\setminus g(B)\\)\uc774\ubbc0\ub85c \\(a\\in g(B)\\)\uc774\ub2e4. \ub610\ud55c \\(g\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \\(g^{-1}(a)\\)\ub294 \ud558\ub098\ub85c \uc815\ud574\uc9c4\ub2e4.<\/p>\n

\uba3c\uc800 \\(h\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(C\\) \uc548\uc5d0\uc11c\ub294 \\(f\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\uace0, \\(A\\setminus C\\)\uc5d0\uc11c\ub294 \\(g^{-1}\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4. \ud55c\ud3b8 \\(a\\in C\\), \\(a’\\notin C\\)\uc774\uace0 \\(h(a)=h(a’)\\)\ub77c\uace0 \uac00\uc815\ud558\uba74
\n\\[
\nf(a)=g^{-1}(a’)
\n\\]
\n\uc774\ubbc0\ub85c
\n\\[
\na’=g(f(a)).
\n\\]
\n\uadf8\ub7ec\ub098 \\(a\\in A_n\\)\uc778 \uc5b4\ub5a4 \\(n\\)\uc774 \uc874\uc7ac\ud558\ubbc0\ub85c
\n\\[
\na’\\in g(f(A_n))=A_{n+1}\\subseteq C
\n\\]
\n\uac00 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(h\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/p>\n

\uc774\uc81c \\(h\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(b\\in B\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(g(b)\\notin C\\)\uc774\uba74
\n\\[
\nh(g(b))=g^{-1}(g(b))=b.
\n\\]
\n\ub9cc\uc57d \\(g(b)\\in C\\)\uc774\uba74 \\(g(b)\\notin A_0\\)\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\ng(b)\\in A_{n+1}=g(f(A_n))
\n\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \\(a\\in A_n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\ng(b)=g(f(a))
\n\\]
\n\uc774\uace0, \\(g\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \\(b=f(a)\\)\uc774\ub2e4. \uc774\ub54c \\(a\\in C\\)\uc774\ubbc0\ub85c
\n\\[
\nh(a)=f(a)=b.
\n\\]
\n\ub530\ub77c\uc11c \\(h\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/p>\n

\uadf8\ub7ec\ubbc0\ub85c \\(h:A\\rightarrow B\\)\ub294 \uc77c\ub300\uc77c \ub300\uc751\uc774\uace0
\n\\[
\n\\lvert A\\rvert=\\lvert B\\rvert
\n\\]
\n\uc774\ub2e4. <\/span><\/p>\n<\/div>\n

\uc9d1\ud569 \\(E\\)\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc744 \\(E\\)\uc758 \uba71\uc9d1\ud569<\/span>(power set)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(P(E)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[
\nP(E)=\\left\\{A\\,\\vert\\,A\\subseteq E\\right\\}
\n\\]
\n\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 0.3.2. (Cantor \uc815\ub9ac)<\/span><\/p>\n

\uc784\uc758\uc758 \uc9d1\ud569 \\(E\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\lvert E\\rvert<\\lvert P(E)\\rvert\n\\]\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\ud568\uc218
\n\\[
\n\\iota:E\\rightarrow P(E),
\n\\qquad
\n\\iota(x)=\\{x\\}
\n\\]
\n\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c
\n\\[
\n\\lvert E\\rvert\\le\\lvert P(E)\\rvert
\n\\]
\n\uc774\ub2e4.<\/p>\n

\ud55c\ud3b8 \\(E\\)\uc5d0\uc11c \\(P(E)\\)\ub85c\uc758 \uc704\ub85c\uc758 \ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0 \uc774\ub97c \\(f:E\\rightarrow P(E)\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc740 \\(E\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.
\n\\[
\nD=\\left\\{x\\in E\\,\\vert\\,x\\notin f(x)\\right\\}.
\n\\]
\n\\(D\\in P(E)\\)\uc774\uace0 \\(f\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\ubbc0\ub85c \\(f(d)=D\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(d\\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(D\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec
\n\\[
\nd\\in D
\n\\quad\\Longleftrightarrow\\quad
\nd\\notin f(d)
\n\\quad\\Longleftrightarrow\\quad
\nd\\notin D
\n\\]
\n\uac00 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(E\\)\uc5d0\uc11c \\(P(E)\\)\ub85c\uc758 \uc704\ub85c\uc758 \ud568\uc218\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba70, \ud2b9\ud788 \ub450 \uc9d1\ud569 \uc0ac\uc774\uc5d0\ub294 \uc77c\ub300\uc77c \ub300\uc751\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[
\n\\lvert E\\rvert<\\lvert P(E)\\rvert\n\\]\n\uc774\ub2e4. <\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 0.3.3.<\/span> \ub2e4\uc74c\uc740 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uc9d1\ud569\uc758 \uae30\uc218\uc5d0 \uad00\ud55c \ud45c\uc900\uc801\uc778 \uacb0\uacfc\uc774\ub2e4. \uc774 \uacb0\uacfc\ub4e4\uc758 \uc99d\uba85\uc740 \uc774 \uae00\uc758 \ubc94\uc704\ub97c \ubc97\uc5b4\ub098\ubbc0\ub85c \uc0dd\ub7b5\ud55c\ub2e4.<\/p>\n

    \n
  1. \\(\\lvert\\mathbb{N}\\rvert=\\lvert\\mathbb{Z}\\rvert=\\lvert\\mathbb{Q}\\rvert.\\)<\/li>\n
  2. \\(\\lvert\\mathbb{N}\\rvert=\\lvert\\mathbb{N}\\times\\mathbb{N}\\rvert.\\)<\/li>\n
  3. \\(\\lvert\\mathbb{N}\\rvert<\\lvert\\mathbb{R}\\rvert.\\)<\/li>\n
  4. \\(\\lvert\\mathbb{R}\\rvert=\\lvert\\mathbb{C}\\rvert.\\)<\/li>\n
  5. \\(\\lvert\\mathbb{R}\\rvert=\\lvert\\mathbb{R}^d\\rvert\\)   (\\(d\\)\ub294 \uc591\uc758 \uc815\uc218).<\/li>\n<\/ol>\n<\/div>\n

    \\(E\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
    \n\\[
    \n\\lvert E\\rvert=\\lvert\\mathbb{N}\\rvert
    \n\\]
    \n\uc774\uba74 \\(E\\)\ub97c \ubb34\ud55c\uac00\uc0b0\uc9d1\ud569<\/span>(countably infinite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(E\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uac70\ub098 \ubb34\ud55c\uac00\uc0b0\uc9d1\ud569\uc774\uba74 \\(E\\)\ub97c \uac00\uc0b0\uc9d1\ud569<\/span>(countable set)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uac00\uc0b0\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc744 \ube44\uac00\uc0b0\uc9d1\ud569<\/span>(uncountable set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c\uc740 \ubaa8\ub450 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.
    \n\\[
    \n\\varnothing,\\qquad
    \n\\left\\{1,\\,2,\\,3\\right\\},\\qquad
    \n\\mathbb{N},\\qquad
    \n\\mathbb{Z},\\qquad
    \n\\mathbb{Q},\\qquad
    \n\\mathbb{N}^2.
    \n\\]
    \n\ubc18\uba74 \\(\\mathbb{R}\\)\uc640 \\(\\mathbb{C}\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n

    <\/p>\n

    \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569\uc758 \uc131\uc9c8<\/h2>\n

    \uc774 \uc808\uc5d0\uc11c\ub294 \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\ub294 \ud1b5\uc0c1\uc801\uc778 \uc9d1\ud569\ub860\uc758 \ud2c0\uc5d0\uc11c \ub2e4\uc74c \uc0ac\uc2e4\uc744 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n

    \n

    \uae30\ubcf8 \uc0ac\uc2e4.<\/span><\/p>\n

    \ubaa8\ub4e0 \ubb34\ud55c\uc9d1\ud569\uc740 \ubb34\ud55c\uac00\uc0b0\uc778 \ubd80\ubd84\uc9d1\ud569\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

    \uc120\ud0dd\uacf5\ub9ac\uc640 \uad00\ub828\ub41c \uc790\uc138\ud55c \ub17c\uc758\ub294 \uc774 \uae00\uc758 \ubc94\uc704\ub97c \ubc97\uc5b4\ub098\ubbc0\ub85c \uc0dd\ub7b5\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 0.3.4.<\/span><\/p>\n

    \\(A,\\) \\(B,\\) \\(C\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

      \n
    1. \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uace0 \\(B\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uba74 \\(\\lvert A\\rvert<\\lvert B\\rvert\\)\uc774\ub2e4.<\/li>\n
    2. \\(\\lvert A\\rvert\\le\\lvert B\\rvert\\)\uc774\uace0 \\(\\lvert B\\rvert\\le\\lvert C\\rvert\\)\uc774\uba74 \\(\\lvert A\\rvert\\le\\lvert C\\rvert\\)\uc774\ub2e4.<\/li>\n
    3. \\(A\\subseteq B\\)\uc774\uace0 \\(B\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \\(A\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n
    4. \\(A\\subseteq B\\)\uc774\uace0 \\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uba74 \\(B\\)\ub3c4 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n
    5. \\(A\\subseteq B\\)\uc774\uace0 \\(B\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\uba74 \\(A\\)\ub3c4 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n
    6. \\(A\\subseteq B\\)\uc774\uace0 \\(A\\)\uac00 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uba74 \\(B\\)\ub3c4 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n
    7. \\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\)\uc640 \uae30\uc218\uac00 \uac19\uc740 \\(A\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc99d\uba85.<\/p>\n

      (1) \\(\\lvert A\\rvert=n\\)\uc774\ub77c\uace0 \ud558\uc790. \uae30\ubcf8 \uc0ac\uc2e4\uc5d0 \uc758\ud558\uc5ec \\(B\\)\ub294 \ubb34\ud55c\uac00\uc0b0\uc778 \ubd80\ubd84\uc9d1\ud569\uc744 \uac00\uc9c0\ubbc0\ub85c \\(A\\)\uc5d0\uc11c \\(B\\)\ub85c\uc758 \uc77c\ub300\uc77c \ud568\uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c
      \n\\[
      \n\\lvert A\\rvert\\le\\lvert B\\rvert.
      \n\\]
      \n\ub9cc\uc57d \\(\\lvert A\\rvert=\\lvert B\\rvert\\)\uc774\uba74 \\(B\\)\ub3c4 \\(n\\)\uac1c\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc720\ud55c\uc9d1\ud569\uc774 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
      \n\\[
      \n\\lvert A\\rvert<\\lvert B\\rvert.\n\\]<\/p>\n

      (2) \uc77c\ub300\uc77c \ud568\uc218 \\(f:A\\rightarrow B\\)\uc640 \\(g:B\\rightarrow C\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(g\\circ f:A\\rightarrow C\\)\ub3c4 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c
      \n\\[
      \n\\lvert A\\rvert\\le\\lvert C\\rvert.
      \n\\]<\/p>\n

      (3) \\(\\lvert B\\rvert=n\\)\uc774\ub77c\uace0 \ud558\uc790. \\(B\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc740 \\(n\\)\uac1c \uc774\ud558\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c0\ubbc0\ub85c \\(A\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      (4) \ub9cc\uc57d \\(B\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 (3)\uc5d0 \uc758\ud558\uc5ec \\(A\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ub2e4. \uc774\ub294 \\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\ub77c\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(B\\)\ub294 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      (5) \uba3c\uc800 \\(B\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 (3)\uc5d0 \uc758\ud558\uc5ec \\(A\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ubbc0\ub85c \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uc774\uc81c \\(B\\)\uac00 \ubb34\ud55c\uac00\uc0b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(B\\)\uc640 \\(\\mathbb{N}\\) \uc0ac\uc774\uc758 \uc77c\ub300\uc77c \ub300\uc751\uc744 \ud558\ub098 \uace0\uc815\ud558\uba74 \\(A\\)\ub294 \\(\\mathbb{N}\\)\uc758 \uc5b4\ub5a4 \ubd80\ubd84\uc9d1\ud569\uacfc \uc77c\ub300\uc77c \ub300\uc751\ud55c\ub2e4. \\(\\mathbb{N}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc740 \uc720\ud55c\uc9d1\ud569\uc774\uac70\ub098, \uc6d0\uc18c\ub4e4\uc744 \uc791\uc740 \uac83\ubd80\ud130 \ucc28\ub840\ub85c \ub098\uc5f4\ud558\uc5ec \\(\\mathbb{N}\\)\uacfc \uc77c\ub300\uc77c \ub300\uc751\uc2dc\ud0ac \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(A\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      (6) \ub9cc\uc57d \\(B\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\uba74 (5)\uc5d0 \uc758\ud558\uc5ec \\(A\\)\ub3c4 \uac00\uc0b0\uc9d1\ud569\uc774 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(B\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      (7) \uba3c\uc800 \\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uae30\ubcf8 \uc0ac\uc2e4\uc5d0 \uc758\ud558\uc5ec \uc11c\ub85c \ub2e4\ub978 \uc6d0\uc18c
      \n\\[
      \na_0,a_1,a_2,\\ldots
      \n\\]
      \n\ub97c \\(A\\)\uc5d0\uc11c \ud0dd\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \\(h:A\\rightarrow A\\setminus\\{a_0\\}\\)\ub97c \uc815\uc758\ud55c\ub2e4.
      \n\\[
      \nh(a_n)=a_{n+1}\\quad(n=0,1,2,\\ldots),
      \n\\]
      \n\uadf8\ub9ac\uace0 \\(x\\notin\\{a_0,a_1,a_2,\\ldots\\}\\)\uc774\uba74 \\(h(x)=x\\)\ub85c \ub193\ub294\ub2e4. \uadf8\ub7ec\uba74 \\(h\\)\ub294 \uc77c\ub300\uc77c \ub300\uc751\uc774\ub2e4. \ub530\ub77c\uc11c \\(A\\)\ub294 \uc790\uc2e0\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \\(A\\setminus\\{a_0\\}\\)\uacfc \uac19\uc740 \uae30\uc218\ub97c \uac00\uc9c4\ub2e4.<\/p>\n

      \uac70\uafb8\ub85c \\(A\\)\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569 \\(D\\)\uac00 \uc874\uc7ac\ud558\uc5ec
      \n\\[
      \n\\lvert A\\rvert=\\lvert D\\rvert
      \n\\]
      \n\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \uc9c4\ubd80\ubd84\uc9d1\ud569 \\(D\\)\ub294 \\(A\\)\ubcf4\ub2e4 \uc801\uc740 \uc218\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c0\ubbc0\ub85c \ub450 \uc9d1\ud569\uc758 \uae30\uc218\uac00 \uac19\uc744 \uc218 \uc5c6\ub2e4. \ub530\ub77c\uc11c \\(A\\)\ub294 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4. <\/span><\/p>\n<\/div>\n

      \ub530\ub77c\uc11c \uc720\ud55c\uc9d1\ud569\uacfc \ub2ec\ub9ac \ubb34\ud55c\uc9d1\ud569\uc740 \uc790\uc2e0\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uacfc \uc77c\ub300\uc77c \ub300\uc751\ud560 \uc218 \uc788\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\ub294 \ud1b5\uc0c1\uc801\uc778 \uc9d1\ud569\ub860\uc5d0\uc11c\ub294 \uc774 \uc131\uc9c8\uc744 \ubb34\ud55c\uc9d1\ud569\uc758 \ud2b9\uc9d5 \uac00\uc6b4\ub370 \ud558\ub098\ub85c \ubcfc \uc218 \uc788\ub2e4.<\/p>\n


      \n<\/p>\n

      <\/p>\n

      <\/p>\n

      \n