{"id":9248,"date":"2025-10-17T19:41:26","date_gmt":"2025-10-17T10:41:26","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9248"},"modified":"2025-10-20T18:48:27","modified_gmt":"2025-10-20T09:48:27","slug":"ch03-algebra-of-classes","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\/","title":{"rendered":"\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0"},"content":{"rendered":"
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\uc9d1\ud569\ub860\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \uae30\ubcf8\uc801\uc778 \uc9d1\ud569\uc758 \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc6b0\ub9ac\ub294 \uc55e\uc5d0\uc11c \ub450 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc5d0 \ub300\ud574\uc11c \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\uac83\uc744 \ud655\uc7a5\ud558\uc5ec \uc9d1\ud569\uc758 \uac1c\uc218\uac00 \uc784\uc758\uc77c \ub54c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc744 \uc815\uc758\ud558\uace0, \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

1. \uc720\ud55c \uac1c \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569<\/h3>\n

\\(A_1,\\, A_2,\\, \\ldots,\\, A_n\\)\uc774 \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub4e4\uc758 \ud569\uc9d1\ud569<\/span>(union)\uc740 \uc774\ub4e4 \uc9d1\ud569 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4. \uc989 \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[A_1 \\cup A_2 \\cup \\cdots \\cup A_n = \\{x \\mid x \\in A_i \\text{ for some } i \\in \\{1,\\, 2,\\, \\ldots,\\, n\\}\\}.\\]
\n\uc774\uac83\uc744 \uac04\ub2e8\ud788 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
\n\\[\\bigcup_{i=1}^{n} A_i \\]
\n\ub9c8\ucc2c\uac00\uc9c0\ub85c \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569<\/span>(intersection)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[A_1 \\cap A_2 \\cap \\cdots \\cap A_n = \\{x \\mid x \\in A_i \\text{ for all } i \\in \\{1,\\, 2,\\, \\ldots,\\, n\\}\\} .\\]
\n\uc774\uac83\uc744 \uac04\ub2e8\ud788 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
\n\\[\\bigcap_{i=1}^{n} A_i\\]
\n\uc608\ub97c \ub4e4\uc5b4, \\(A_1 = \\{1,\\, 2,\\, 3\\}\\), \\(A_2 = \\{2,\\, 3,\\, 4\\}\\), \\(A_3 = \\{3,\\, 4,\\, 5\\}\\)\uc77c \ub54c, \uc138 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc740 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{aligned}
\n\\bigcup_{i=1}^{3} A_i = \\{1,\\, 2,\\, 3,\\, 4,\\, 5\\} \\quad \\text{\uadf8\ub9ac\uace0} \\quad
\n\\bigcap_{i=1}^{3} A_i = \\{3\\} .
\n\\end{aligned}\\]<\/p>\n

\n

\ubb38\uc81c 3.1.<\/span>
\n\ub2e4\uc74c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

    \n
  1. \\(A_1 = \\left\\{ 2,\\,4,\\,6\\right\\}\\), \\(A_2 = \\left\\{ 2,\\,4,\\,8 \\right\\}\\), \\(A_3 = \\left\\{ 4,\\,6,\\,13 \\right\\}\\)\uc77c \ub54c \\(\\bigcup_{i=1}^{3} A_i\\)\uc640 \\(\\bigcap_{i=1}^{3} A_i .\\)<\/li>\n
  2. \\(B_j = \\left\\{ j,\\, j+1 ,\\, j+2 ,\\, j+3 \\right\\}\\)\uc77c \ub54c \\(\\bigcup_{j=1}^{3} B_j\\)\uc640 \\(\\bigcap_{j=1}^{3} B_j .\\)<\/li>\n
  3. \\(C_k = \\left\\{ n \\,\\vert\\, n\\text{\uc740}\\,\\,k\\text{\uc758 \ubc30\uc218\uc778 \uc790\uc5f0\uc218}\\right\\}\\)\uc77c \ub54c \\(\\bigcup_{k=3}^{6} C_k\\)\uc640 \\(\\bigcap_{k=3}^{7} C_k\\).<\/li>\n
  4. \\(D_k = \\left\\{ n \\,\\vert\\, n\\text{\uc740}\\,\\,k\\text{\uc758 \uc57d\uc218\uc778 \uc790\uc5f0\uc218}\\right\\}\\)\uc77c \ub54c \\(\\bigcup_{k=3}^{6} D_k\\)\uc640 \\(\\bigcap_{k=4}^{6} D_{3k}\\).<\/li>\n
  5. \\(E_{(i,\\,j)} = \\left\\{ n\\,\\vert\\, n\\text{\uc740}\\,\\, i \\le n \\le j\\text{\uc778 \uc815\uc218}\\right\\}\\)\uc77c \ub54c \\(\\bigcup_{i=1}^{3} \\left( \\bigcap_{j=5}^{7} E_{(i,\\,j)} \\right)\\).<\/li>\n<\/ol>\n<\/div>\n

    2. \uc9d1\ud569\uc758 \uac1c\uc218\uac00 \uac00\uc0b0\uc77c \ub54c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569<\/h3>\n

    \uac1c\uc218\uac00 \uac00\uc0b0 \ubb34\ud55c[\uc720\ud55c\uc774\uac70\ub098 \ub610\ub294 \uc790\uc5f0\uc218 \ub9cc\ud07c \ub9ce\uc740 \ubb34\ud55c]\uc778 \uc9d1\ud569 \\(A_1,\\, A_2,\\, A_3,\\, \\ldots\\)\uc5d0 \ub300\ud574 \uc0dd\uac01\ud574\ubcf4\uc790. \uc774\ub4e4\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc740 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
    \n\\[\\begin{aligned}
    \n\\bigcup_{i=1}^{\\infty} A_i &= \\{x \\mid x \\in A_i \\text{ for some } i \\in \\mathbb{Z}^+\\} ,\\\\
    \n\\bigcap_{i=1}^{\\infty} A_i &= \\{x \\mid x \\in A_i \\text{ for all } i \\in \\mathbb{Z}^+\\} .
    \n\\end{aligned}\\]
    \n\uc608\ub97c \ub4e4\uc5b4, \\(A_n = \\{n,\\, n+1,\\, n+2,\\, \\ldots\\}\\)\ub77c\uace0 \ud558\uba74, \\(A_n\\)\ub4e4\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\bigcup_{n=1}^{\\infty} A_n = \\mathbb{Z}^+ \\quad \\text{\uadf8\ub9ac\uace0} \\quad \\bigcap_{n=1}^{\\infty} A_n = \\varnothing .\\]<\/p>\n

    \n

    \ubb38\uc81c 3.2.<\/span>
    \n\ub2e4\uc74c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \\(A_n = [n-3 ,\\, n+3]\\)\uc77c \ub54c \\(\\bigcup_{n=1}^{\\infty} A_n\\)\uacfc \\(\\bigcap_{n=1}^{\\infty} A_n\\).<\/li>\n
    2. \\(B_k = \\left[ 2-\\frac{1}{k} ,\\, 4+ \\frac{1}{k} \\right]\\)\uc77c \ub54c \\(\\bigcup_{k=1}^{\\infty} B_k\\)\uc640 \\(\\bigcap_{k=1}^{\\infty} B_k\\).<\/li>\n
    3. \\(C_k = \\left[ 2+\\frac{1}{k} ,\\, 4- \\frac{1}{k} \\right]\\)\uc77c \ub54c \\(\\bigcup_{k=1}^{\\infty} C_k\\)\uc640 \\(\\bigcap_{k=1}^{\\infty} C_k\\).<\/li>\n
    4. \\(D_k = \\left[ 2-\\frac{1}{k} ,\\, 4+ \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\bigcup_{k=1}^{\\infty} D_k\\)\uc640 \\(\\bigcap_{k=1}^{\\infty} D_k\\).<\/li>\n
    5. \\(E_k = \\left[ 2+\\frac{1}{k} ,\\, 4- \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\bigcup_{k=1}^{\\infty} E_k\\)\uc640 \\(\\bigcap_{k=1}^{\\infty} E_k\\).<\/li>\n
    6. \\(G_k = \\left( 2 ,\\, 2+ \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\bigcup_{k=1}^{\\infty} G_k\\)\uc640 \\(\\bigcap_{k=1}^{\\infty} G_k\\).<\/li>\n<\/ol>\n<\/div>\n

      3. \uc9d1\ud569\uc758 \uac1c\uc218\uac00 \uc784\uc758\uc77c \ub54c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569<\/h3>\n

      \ub354 \uc77c\ubc18\uc801\uc73c\ub85c, \ucca8\uc790\uc9d1\ud569<\/span>(index set) \\(I\\)\uc5d0 \uc758\ud574 \ucca8\uc790\uac00 \ub9e4\uaca8\uc9c4 \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i \\in I}\\)\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. [\\(\\{A_i\\}_{i \\in I}\\)\ub97c \\(\\{A_i \\mid i \\in I\\}\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.] \uc5ec\uae30\uc11c \\(I\\)\ub294 \uc720\ud55c\uc9d1\ud569\uc77c \uc218\ub3c4 \uc788\uace0, \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc77c \uc218\ub3c4 \uc788\uc73c\uba70, \ube44\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc77c \uc218\ub3c4 \uc788\ub2e4.<\/p>\n

      \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i \\in I}\\)\uc758 \ud569\uc9d1\ud569<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
      \n\\[\\bigcup_{i \\in I} A_i = \\{x \\mid \\exists i \\in I : x \\in A_i\\} .\\]
      \n\uc989, \uc774 \ud569\uc9d1\ud569\uc740 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \\(A_i\\)\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

      \ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i \\in I}\\)\uc758 \uad50\uc9d1\ud569<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
      \n\\[\\bigcap_{i \\in I} A_i = \\{x \\mid \\forall i \\in I : x \\in A_i\\} .\\]
      \n\uc989, \uc774 \uad50\uc9d1\ud569\uc740 \ubaa8\ub4e0 \\(A_i\\)\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

      4. \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569 \uc5f0\uc0b0\uc758 \uc131\uc9c8<\/h3>\n

      \uc784\uc758\uc758 \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i \\in I}\\)\uc640 \uc9d1\ud569 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n