{"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":"2021-08-01T19:44:14","modified_gmt":"2021-08-01T10:44:14","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\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ub41c\ub2e4.
\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&\\,\\,\\,\\vdots
\n\\end{align}\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(n\\)\uc774 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\ub77c\uba74, \\(n\\)\uc740 \\(n\\)\uac1c\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774\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 \\(\\lvert A \\rvert = \\lvert B \\rvert\\) \ub610\ub294 \\(A \\approx B\\)\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 \\(\\lvert E \\rvert = \\lvert k \\rvert\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(E\\)\ub97c \uc720\ud55c\uc9d1\ud569<\/span>(finite set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uacbd\uc6b0 \\(E\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub294 \\(k\\)\uc774\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c \\(n(E) = k\\) \ub610\ub294 \\(\\lvert E \\rvert = k\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ub9cc\uc57d \\(\\lvert E \\rvert = k \\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uace0 \uc74c\uc774 \uc544\ub2cc \uc815\uc218 \\(k\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\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\\)\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \uc77c\ub300\uc77c \ud568\uc218\uac00 \uc874\uc7ac\ud558\uba74 \uc774\uac83\uc744 \uae30\ud638\ub85c \\(\\lvert A \\rvert \\le \\lvert B \\rvert\\)\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 \uc774\uac83\uc744 \uae30\ud638\ub85c \\(\\lvert A \\rvert < \\lvert B \\rvert\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub85c\uc368 \uc9d1\ud569\uc758 \uc6d0\uc18c\uc758 \uc218\uac00 \ubb34\ud55c\uc774 \ub9ce\uc744 \ub54c\uc5d0\ub3c4 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud560 \uc218 \uc788\uac8c \ub418\uc5c8\ub2e4.<\/p>\n

\ub2e4\uc74c \uc815\ub9ac\ub294 \ubb34\ud55c\uc9d1\ud569\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \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 \\(\\lvert A \\rvert \\le \\lvert B \\rvert\\)\uc774\uace0 \\(\\lvert B \\rvert \\le \\lvert A \\rvert\\)\uc774\uba74, \\(\\lvert A \\rvert = \\lvert B \\rvert\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 0.3.2.<\/span> \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\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
  6. \\(\\lvert E \\rvert < \\lvert P(E) \\rvert \\)   (\\(E\\)\ub294 \uc784\uc758\uc758 \uc9d1\ud569, \\(P(E)\\)\ub294 \\(E\\)\uc758 \uba71\uc9d1\ud569).<\/li>\n<\/ol>\n<\/div>\n

    \\(E\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uac70\ub098 \\(\\lvert E \\rvert = \\lvert \\mathbb{N} \\rvert\\)\uc774\uba74, \\(E\\)\ub97c \uac00\uc0b0\uc9d1\ud569<\/span>(countable set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uac00\uc0b0\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc744 \ube44\uac00\uc0b0\uc9d1\ud569<\/span>(uncountable set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub2e4\uc74c\uc740 \ubaa8\ub450 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.
    \n\\[\\varnothing ,\\quad \\left\\{ 1,\\,2,\\,3 \\right\\} ,\\quad \\mathbb{N} ,\\quad \\mathbb{Z} ,\\quad \\mathbb{Q} ,\\quad \\mathbb{N}^2 .\\]
    \n\ubc18\uba74 \\(\\mathbb{R} \\)\uc640 \\(\\mathbb{C}\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.\n<\/p>\n

    <\/p>\n

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

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