{"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":"2026-09-27T11:43:42","modified_gmt":"2026-09-27T02:43:42","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":"
<\/p>\n

<\/p>\n

\uc55e \uc7a5\uc5d0\uc11c\ub294 \ub450 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569 \ubc0f \uae30\ubcf8\uc801\uc778 \uc9d1\ud569 \uc5f0\uc0b0\uc758 \ubc95\uce59\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub97c \uc9d1\ud569\uc871\uc73c\ub85c \ud655\uc7a5\ud558\uc5ec \uc720\ud55c \uac1c \ub610\ub294 \uc784\uc758 \uac1c\uc758 \uc9d1\ud569\uc5d0 \ub300\ud55c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc744 \uc815\uc758\ud558\uace0, \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774\uc5b4\uc11c \ub450 \uc9d1\ud569\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uacfc \uc77c\ubc18\ud654\ub41c \ub370\uce74\ub974\ud2b8 \uacf1\uc744 \ub2e4\ub8ec\ub2e4.<\/p>\n

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

\uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(A_1,\\, A_2,\\, \\ldots,\\, A_n\\)\uc774 \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub4e4\uc758 \ud569\uc9d1\ud569\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\\[A_1 \\cup A_2 \\cup \\cdots \\cup A_n
\n= \\{x \\mid \\text{\uc5b4\ub5a4 } i \\in \\{1,\\,2,\\,\\ldots,\\,n\\}\\text{\uc5d0 \ub300\ud558\uc5ec } x \\in A_i\\}\\]
\n\ub85c \uc815\uc758\ud558\uace0, \uc774\ub97c \uac04\ub2e8\ud788
\n\\[\\bigcup_{i=1}^{n} A_i\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc744
\n\\[A_1 \\cap A_2 \\cap \\cdots \\cap A_n
\n= \\{x \\mid \\text{\ubaa8\ub4e0 } i \\in \\{1,\\,2,\\,\\ldots,\\,n\\}\\text{\uc5d0 \ub300\ud558\uc5ec } x \\in A_i\\}\\]
\n\ub85c \uc815\uc758\ud558\uace0, \uc774\ub97c \uac04\ub2e8\ud788
\n\\[\\bigcap_{i=1}^{n} A_i\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \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\\[
\n\\bigcup_{i=1}^{3} A_i = \\{1,\\,2,\\,3,\\,4,\\,5\\},\\quad
\n\\bigcap_{i=1}^{3} A_i = \\{3\\}.
\n\\]<\/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 \\(\\displaystyle\\bigcup_{i=1}^{3} A_i\\)\uc640 \\(\\displaystyle\\bigcap_{i=1}^{3} A_i .\\)<\/li>\n
  2. \\(B_j = \\left\\{ j,\\, j+1 ,\\, j+2 ,\\, j+3 \\right\\}\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{j=1}^{3} B_j\\)\uc640 \\(\\displaystyle\\bigcap_{j=1}^{3} B_j .\\)<\/li>\n
  3. \\(C_k = \\left\\{ n \\,\\vert\\, n\\text{\uc740 }k\\text{\uc758 \ubc30\uc218\uc778 \uc591\uc758 \uc815\uc218}\\right\\}\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=3}^{6} C_k\\)\uc640 \\(\\displaystyle\\bigcap_{k=3}^{7} C_k\\).<\/li>\n
  4. \\(D_k = \\left\\{ n \\,\\vert\\, n\\text{\uc740 }k\\text{\uc758 \uc591\uc758 \uc57d\uc218}\\right\\}\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=3}^{6} D_k\\)\uc640 \\(\\displaystyle\\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 \\(\\displaystyle\\bigcup_{i=1}^{3} \\left( \\bigcap_{j=5}^{7} E_{(i,\\,j)} \\right)\\).<\/li>\n<\/ol>\n<\/div>\n

    2. \uc790\uc5f0\uc218\ub85c \ucca8\uc790\uac00 \ub9e4\uaca8\uc9c4 \uc9d1\ud569\uc871\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569<\/h3>\n

    \uc591\uc758 \uc815\uc218 \\(1,\\,2,\\,3,\\,\\ldots\\)\ub85c \ucca8\uc790\uac00 \ub9e4\uaca8\uc9c4 \uc9d1\ud569\uc871 \\(A_1,\\,A_2,\\,A_3,\\,\\ldots\\)\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790. \uac00\uc0b0\uc131\uacfc \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc740 \ub4a4\uc5d0\uc11c \uc815\uc2dd\uc73c\ub85c \uc815\uc758\ud558\ubbc0\ub85c, \uc5ec\uae30\uc11c\ub294 \uc790\uc5f0\uc218\ub85c \ucca8\uc790\uac00 \ub9e4\uaca8\uc9c4 \uc9d1\ud569\uc871\ub9cc \ub2e4\ub8ec\ub2e4. \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
    \n&= \\{x \\mid \\text{\uc5b4\ub5a4 } i \\in \\mathbb{Z}^+\\text{\uc5d0 \ub300\ud558\uc5ec } x \\in A_i\\},\\\\
    \n\\bigcap_{i=1}^{\\infty} A_i
    \n&= \\{x \\mid \\text{\ubaa8\ub4e0 } i \\in \\mathbb{Z}^+\\text{\uc5d0 \ub300\ud558\uc5ec } x \\in A_i\\}.
    \n\\end{aligned}\\]<\/p>\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 \\(\\displaystyle\\bigcup_{n=1}^{\\infty} A_n\\)\uacfc \\(\\displaystyle\\bigcap_{n=1}^{\\infty} A_n\\).<\/li>\n
    2. \\(B_k = \\left[ 2-\\frac{1}{k} ,\\, 4+ \\frac{1}{k} \\right]\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=1}^{\\infty} B_k\\)\uc640 \\(\\displaystyle\\bigcap_{k=1}^{\\infty} B_k\\).<\/li>\n
    3. \\(C_k = \\left[ 2+\\frac{1}{k} ,\\, 4- \\frac{1}{k} \\right]\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=1}^{\\infty} C_k\\)\uc640 \\(\\displaystyle\\bigcap_{k=1}^{\\infty} C_k\\).<\/li>\n
    4. \\(D_k = \\left[ 2-\\frac{1}{k} ,\\, 4+ \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=1}^{\\infty} D_k\\)\uc640 \\(\\displaystyle\\bigcap_{k=1}^{\\infty} D_k\\).<\/li>\n
    5. \\(E_k = \\left[ 2+\\frac{1}{k} ,\\, 4- \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=1}^{\\infty} E_k\\)\uc640 \\(\\displaystyle\\bigcap_{k=1}^{\\infty} E_k\\).<\/li>\n
    6. \\(G_k = \\left( 2 ,\\, 2+ \\frac{1}{k} \\right)\\)\uc77c \ub54c \\(\\displaystyle\\bigcup_{k=1}^{\\infty} G_k\\)\uc640 \\(\\displaystyle\\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\\)\uc758 \uac01 \uc6d0\uc18c \\(i\\)\uc5d0 \uc9d1\ud569 \\(A_i\\)\uac00 \ub300\uc751\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \uc774\uc640 \uac19\uc774 \ucca8\uc790\uac00 \ub9e4\uaca8\uc9c4 \uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784 \\(\\{A_i\\}_{i\\in I}\\)\ub97c \uc9d1\ud569\uc871<\/span>(family of sets)\uc774\ub77c\uace0 \ud55c\ub2e4. [\ubb38\ub9e5\uc5d0 \ub530\ub77c \uc9d1\ud569\uc871\uc5d0 \ub098\ud0c0\ub098\ub294 \uc9d1\ud569\ub4e4\ub9cc \ud45c\uc2dc\ud560 \ub54c \\(\\{A_i \\mid i\\in I\\}\\)\ub77c\uace0 \uc4f0\uae30\ub3c4 \ud55c\ub2e4. \ub2e4\ub9cc \uc774 \ud45c\uae30\ub294 \uc11c\ub85c \ub2e4\ub978 \ucca8\uc790\uac00 \uac19\uc740 \uc9d1\ud569\uc744 \uac00\ub9ac\ud0a4\ub294 \uc815\ubcf4\ub97c \uad6c\ubcc4\ud558\uc9c0 \uc54a\ub294\ub2e4.] \uc5ec\uae30\uc11c\ub294 \uba3c\uc800 \\(I\\ne\\varnothing\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud55c\ub2e4.<\/p>\n

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

      \n

      \ubb38\uc81c 3.3.<\/span>
      \n\uc5f4\ub9b0\uad6c\uac04\uc73c\ub85c \uc815\uc758\ub41c \ucca8\uc790\uc9d1\ud569 \\(I=(0,\\,\\infty)\\)\uc5d0 \ub300\ud558\uc5ec, \uac01 \\(t\\in I\\)\uc5d0 \ub300\ud574
      \n\\[
      \nA_t=(-t,\\,t),\\quad B_t=(-\\infty,\\,t)
      \n\\]
      \n\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

        \n
      1. \\(\\displaystyle\\bigcup_{t\\in I}A_t\\)<\/li>\n
      2. \\(\\displaystyle\\bigcap_{t\\in I}A_t\\)<\/li>\n
      3. \\(\\displaystyle\\bigcup_{t\\in I}B_t\\)<\/li>\n
      4. \\(\\displaystyle\\bigcap_{t\\in I}B_t\\)<\/li>\n<\/ol>\n<\/div>\n

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

        \uc815\ub9ac 3.1. (\uc9d1\ud569\uc871\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc758 \ubc95\uce59)<\/span><\/p>\n

        \uc804\uccb4\uc9d1\ud569 \\(U\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0, \\(I\\ne\\varnothing\\), \\(A_i\\subseteq U\\) \\((i\\in I)\\), \\(B\\subseteq U\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n
        1. \ubd84\ubc30\ubc95\uce59<\/span>:
          \n\\[\\begin{aligned}
          \nB \\cap \\left(\\bigcup_{i\\in I} A_i\\right)
          \n&= \\bigcup_{i\\in I}(B\\cap A_i),\\\\
          \nB \\cup \\left(\\bigcap_{i\\in I} A_i\\right)
          \n&= \\bigcap_{i\\in I}(B\\cup A_i).
          \n\\end{aligned}\\]\n<\/li>\n
        2. \ub4dc \ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span>:
          \n\\[\\begin{aligned}
          \n\\left(\\bigcup_{i\\in I} A_i\\right)^c
          \n&= \\bigcap_{i\\in I} A_i^c,\\\\
          \n\\left(\\bigcap_{i\\in I} A_i\\right)^c
          \n&= \\bigcup_{i\\in I} A_i^c.
          \n\\end{aligned}\\]
          \n\uc5ec\uae30\uc11c \uc5ec\uc9d1\ud569\uc740 \ubaa8\ub450 \\(U\\)\uc5d0 \ub300\ud55c \uc5ec\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 3.4.<\/span>
          \n\uc815\ub9ac 3.1\uc758 \ubd84\ubc30\ubc95\uce59\uacfc \ub4dc \ubaa8\ub974\uac04\uc758 \ubc95\uce59\uc744 \uc99d\uba85\ud558\uc2dc\uc624. (\ub2e8, \\(I\\ne\\varnothing\\)\uc778 \uac83\uc73c\ub85c \uac00\uc815\ud55c\ub2e4.)<\/p>\n<\/div>\n

          5. \ucca8\uc790\uc9d1\ud569\uc774 \uacf5\uc9d1\ud569\uc778 \uacbd\uc6b0<\/h3>\n

          \ucca8\uc790\uc9d1\ud569\uc774 \uacf5\uc9d1\ud569\uc778 \uacbd\uc6b0, \uc989 \\(I=\\varnothing\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. \uad50\uc9d1\ud569\uae4c\uc9c0 \ud558\ub098\uc758 \uc9d1\ud569\uc73c\ub85c \ub2e4\ub8e8\uae30 \uc704\ud574 \uc804\uccb4\uc9d1\ud569 \\(U\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \ubaa8\ub4e0 \\(A_i\\)\uac00 \\(U\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. [\uc5ec\uae30\uc11c \\(U\\)\ub294 \ud604\uc7ac \ub17c\uc758\uc758 \ub300\uc0c1 \uc601\uc5ed\uc73c\ub85c \uace0\uc815\uc2dc\ud0a8 \ud558\ub098\uc758 \uc9d1\ud569\uc774\uba70, \ubaa8\ub4e0 \uc9d1\ud569\uc744 \uc6d0\uc18c\ub85c \uac00\uc9c0\ub294 ‘\ubaa8\ub4e0 \uc9d1\ud569\uc758 \uc9d1\ud569’\uc744 \ub73b\ud558\uc9c0 \uc54a\ub294\ub2e4.]<\/p>\n

          \ud569\uc9d1\ud569\uc758 \uc815\uc758\uc5d0\uc11c \\(\\bigcup_{i\\in\\varnothing}A_i\\)\uc5d0 \uc18d\ud558\ub824\uba74 \uc5b4\ub5a4 \\(i\\in\\varnothing\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in A_i\\)\uc774\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub098 \uadf8\ub7f0 \\(i\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c
          \n\\[\\bigcup_{i\\in\\varnothing}A_i=\\varnothing\\]
          \n\ub85c \ub454\ub2e4.<\/p>\n

          \ud55c\ud3b8 \ubaa8\ub4e0 \\(x\\in U\\)\uc5d0 \ub300\ud558\uc5ec “\ubaa8\ub4e0 \\(i\\in\\varnothing\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in A_i\\)\uc774\ub2e4”\ub77c\ub294 \uc804\uce6d\uba85\uc81c\ub294 \uacf5\ud5c8\ud558\uac8c \ucc38\uc774\ub2e4. \ub530\ub77c\uc11c
          \n\\[\\bigcap_{i\\in\\varnothing}A_i=U\\]
          \n\ub85c \ub454\ub2e4. \uc774\ub294 \ucca8\uc790\uc9d1\ud569\uc744 \uc904\uc600\uc744 \ub54c \ud569\uc9d1\ud569\uc740 \uc791\uc544\uc9c0\uace0 \uad50\uc9d1\ud569\uc740 \ucee4\uc9c0\ub294 \uc131\uc9c8\uacfc\ub3c4 \uc77c\uce58\ud55c\ub2e4. \uc2e4\uc81c\ub85c \\(J\\subseteq I\\)\uc774\uba74
          \n\\[\\bigcup_{j\\in J}A_j\\subseteq\\bigcup_{i\\in I}A_i
          \n\\quad\\text{\uc774\uace0}\\quad
          \n\\bigcap_{i\\in I}A_i\\subseteq\\bigcap_{j\\in J}A_j.\\]
          \n\ub530\ub77c\uc11c \ucca8\uc790\uc9d1\ud569\uc73c\ub85c\uc11c \uacf5\uc9d1\ud569\uc744 \ud5c8\uc6a9\ud558\uba74 \ud569\uc9d1\ud569\uc758 \ucd5c\uc18c\uac12\uc740 \\(\\varnothing\\), \uad50\uc9d1\ud569\uc758 \ucd5c\ub300\uac12\uc740 \\(U\\)\uac00 \ub41c\ub2e4. \uc774 \uc57d\uc18d\uc744 \uc0ac\uc6a9\ud558\uba74 \uc815\ub9ac 3.1\uc758 \ub4f1\uc2dd\ub4e4\ub3c4 \\(I=\\varnothing\\)\uc778 \uacbd\uc6b0\uae4c\uc9c0 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 3.5.<\/span>
          \n\uc804\uccb4\uc9d1\ud569 \\(U\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \ucca8\uc790\uc9d1\ud569\uc744 \\(I=\\varnothing\\)\uc774\ub77c \ud558\uc790. \ub610\ud55c \\(B\\subseteq U\\)\ub77c \ud558\uc790.<\/p>\n

            \n
          1. \\(\\displaystyle\\bigcup_{i\\in I}A_i\\)\uc640 \\(\\displaystyle\\bigcap_{i\\in I}A_i\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n
          2. \ub2e4\uc74c \ub124 \uc9d1\ud569\uc744 \uac01\uac01 \uad6c\ud558\uace0, \\(I=\\varnothing\\)\uc77c \ub54c\uc5d0\ub3c4 \ubd84\ubc30\ubc95\uce59\uc774 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud558\uc2dc\uc624.
            \n\\[
            \nB\\cap\\left(\\bigcup_{i\\in I}A_i\\right),\\quad
            \n\\bigcup_{i\\in I}(B\\cap A_i),\\quad
            \nB\\cup\\left(\\bigcap_{i\\in I}A_i\\right),\\quad
            \n\\bigcap_{i\\in I}(B\\cup A_i).
            \n\\]\n<\/li>\n
          3. \ub4dc \ubaa8\ub974\uac04\uc758 \ubc95\uce59
            \n\\[
            \n\\left(\\bigcup_{i\\in I}A_i\\right)^c=\\bigcap_{i\\in I}A_i^c,
            \n\\quad
            \n\\left(\\bigcap_{i\\in I}A_i\\right)^c=\\bigcup_{i\\in I}A_i^c
            \n\\]
            \n\ub3c4 \uc9c1\uc811 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

            6. \ub370\uce74\ub974\ud2b8 \uacf1<\/h3>\n

            \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1<\/span>(Cartesian product) \\(A\\times B\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc21c\uc11c\uc30d\ub4e4\uc758 \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.
            \n\\[A\\times B=\\{(a,\\,b)\\mid a\\in A,\\, b\\in B\\}.\\]
            \n\uc21c\uc11c\uc30d\uc5d0\uc11c\ub294 \uc131\ubd84\uc758 \uc21c\uc11c\uac00 \uc911\uc694\ud558\uba70,
            \n\\[(a,b)=(a’,b’)\\quad\\Longleftrightarrow\\quad a=a’\\text{\uc774\uace0 }b=b’\\]
            \n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

            \uc608\ub97c \ub4e4\uc5b4, \\(A = \\{1,\\, 2\\}\\), \\(B = \\{x,\\, y,\\, z\\}\\)\uc77c \ub54c,
            \n\\[A \\times B = \\{(1,\\,x),\\, (1,\\,y),\\, (1,\\,z),\\, (2,\\,x),\\, (2,\\,y),\\, (2,\\,z)\\}.\\]<\/p>\n

            \uc720\ud55c \uac1c\uc758 \uc9d1\ud569 \\(A_1,\\, A_2,\\, \\ldots,\\, A_n\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
            \n\\[A_1\\times A_2\\times\\cdots\\times A_n
            \n=\\{(a_1,\\,a_2,\\,\\ldots,\\,a_n)\\mid a_i\\in A_i\\text{\uc774\uace0 }i=1,\\,\\ldots,\\,n\\}.\\]
            \n\uc774\uac83\uc744 \uac04\ub2e8\ud788 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.
            \n\\[\\prod_{i=1}^{n} A_i \\]<\/p>\n

            \n

            \ubb38\uc81c 3.6.<\/span>
            \n\uc9d1\ud569
            \n\\[
            \nA=\\{1,\\,2\\},\\quad B=\\{a,\\,b\\},\\quad C=\\{0,\\,1,\\,2\\}
            \n\\]
            \n\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \uad6c\ud558\uac70\ub098 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

              \n
            1. \\(A\\times B\\)\uc640 \\(B\\times A\\)\ub97c \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
            2. \\(A\\times B=B\\times A\\)\uc778\uc9c0 \ud310\ub2e8\ud558\uace0 \uadf8 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
            3. \\(A\\times B\\times C\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uace0, \uccab \ubc88\uc9f8 \uc131\ubd84\uc774 \\(2\\)\uc778 \uc6d0\uc18c\ub97c \ubaa8\ub450 \uc4f0\uc2dc\uc624.<\/li>\n
            4. \\(A\\times\\varnothing\\)\uacfc \\(\\varnothing\\times B\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
              \n

              \ubb38\uc81c 3.7.<\/span>
              \n\\(A_1,\\,\\ldots,\\,A_n\\)\uc774 \uc720\ud55c\uc9d1\ud569\uc77c \ub54c \\(n(A_1),\\,\\ldots,\\,n(A_n)\\)\uacfc \\(n\\!\\left(\\prod_{i=1}^{n}A_i\\right)\\) \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 3.8.<\/span>
              \n\\(A\\), \\(B\\), \\(C\\)\uac00 \uc9d1\ud569\uc77c \ub54c, \uc138 \uc9d1\ud569
              \n\\[A\\times B\\times C,\\quad (A\\times B)\\times C,\\quad A\\times(B\\times C)\\]
              \n\uac00 \uc77c\ubc18\uc801\uc73c\ub85c \uac19\uc740 \uc9d1\ud569\uc778\uc9c0 \ud655\uc778\ud558\uc2dc\uc624. \ub610\ud55c \uc774 \uc138 \uc9d1\ud569 \uc0ac\uc774\uc758 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c \ub300\uc751\uc744 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              7. \ud568\uc218\ub97c \uc0ac\uc6a9\ud55c \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uc815\uc758<\/h3>\n

              \ud568\uc218\uc758 \uc815\ud655\ud55c \uc815\uc758\ub294 \ub2e4\uc74c \uc7a5\uc5d0\uc11c \ub2e4\ub8ec\ub2e4. \uc5ec\uae30\uc11c\ub294 \uac01 \uc785\ub825\uc5d0 \ud558\ub098\uc758 \uac12\uc744 \ub300\uc751\uc2dc\ud0a4\ub294 \ub300\uc0c1\uc73c\ub85c \ud568\uc218\ub97c \uc9c1\uad00\uc801\uc73c\ub85c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n

              \\(n\\)-\uc21c\uc11c\uc30d \\((a_1,\\,a_2,\\,\\ldots,\\,a_n)\\)\uc740 \uc815\uc758\uc5ed\uc774 \\(\\{1,\\,2,\\,\\ldots,\\,n\\}\\)\uc778 \ud568\uc218\uc640 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub300\uc751\uc2dc\ud0ac \uc218 \uc788\ub2e4. \uc989,
              \n\\[f\\colon\\{1,\\,2,\\,\\ldots,\\,n\\}\\to\\bigcup_{i=1}^{n}A_i,\\quad f(i)=a_i\\in A_i\\]
              \n\uc778 \ud568\uc218 \\(f\\)\ub97c \uc21c\uc11c\uc30d \\((a_1,\\,\\ldots,\\,a_n)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4.<\/p>\n

              \uc774\ub7ec\ud55c \uad00\uc810\uc740 \uc784\uc758\uc758 \ucca8\uc790\uc9d1\ud569\uc5d0 \ub300\ud55c \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uc815\uc758\ub85c \uc790\uc5f0\uc2a4\ub7fd\uac8c \ud655\uc7a5\ub41c\ub2e4. \ucca8\uc790\uc9d1\ud569 \\(I\\)\uc640 \uc9d1\ud569\uc871 \\(\\{A_i\\}_{i\\in I}\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[\\prod_{i\\in I}A_i
              \n=\\left\\{f\\colon I\\to\\bigcup_{i\\in I}A_i\\,\\middle|\\, \\text{\ubaa8\ub4e0 }i\\in I\\text{\uc5d0 \ub300\ud558\uc5ec }f(i)\\in A_i\\right\\}\\]
              \n\ub85c \uc815\uc758\ud55c\ub2e4. \\(I=\\{1,\\ldots,n\\}\\)\uc774\uba74 \uc774\ub294 \uc704\uc758 \\(n\\)-\uc21c\uc11c\uc30d \ud45c\ud604\uacfc \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub300\uc751\ud55c\ub2e4.<\/p>\n

              \n

              \ubb38\uc81c 3.9.<\/span>
              \n\\(I=\\{1,\\,2,\\,3\\}\\)\uc774\uace0
              \n\\[
              \nA_1=\\{a,\\,b\\},\\quad A_2=\\{0,\\,1\\},\\quad A_3=\\{\\ast\\}
              \n\\]
              \n\ub77c\uace0 \ud558\uc790.<\/p>\n

                \n
              1. \\(\\displaystyle\\prod_{i\\in I}A_i\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \\(3\\)-\uc21c\uc11c\uc30d\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
              2. (1)\uc758 \uac01 \uc21c\uc11c\uc30d \\((x_1,\\,x_2,\\,x_3)\\)\uc744 \ud568\uc218 \\(f\\colon I\\to A_1\\cup A_2\\cup A_3\\)\ub85c \ubcf4\uc558\uc744 \ub54c \\(f(1)\\), \\(f(2)\\), \\(f(3)\\)\uc744 \uc5b4\ub5bb\uac8c \uc815\ud574\uc57c \ud558\ub294\uc9c0 \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
              3. \\(\\displaystyle n\\!\\left(\\prod_{i\\in I}A_i\\right)\\)\uc744 \uad6c\ud558\uace0, \\(n(A_1)n(A_2)n(A_3)\\)\uacfc \ube44\uad50\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

                \\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc77c \ub54c, \\(B\\)\uc5d0\uc11c \\(A\\)\ub85c \uac00\ub294 \ubaa8\ub4e0 \ud568\uc218\uc758 \uc9d1\ud569\uc744 \\(A^B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud55c\ud3b8 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc758 \\(n\\)-\uc911 \ub370\uce74\ub974\ud2b8 \uacf1
                \n\\[\\underbrace{A\\times\\cdots\\times A}_{n\\text{\uac1c}}\\]
                \n\ub97c \\(A^n\\)\uc73c\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \ud2b9\ud788 \\(I=\\{1,\\,\\ldots,\\,n\\}\\)\uc77c \ub54c \\(A^n\\)\uacfc \\(A^I\\)\ub294 \ubb38\uc81c 3.10\uc5d0\uc11c \uc124\uba85\ud558\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \ub300\uc751\uc73c\ub85c \uc5f0\uacb0\ub41c\ub2e4.<\/p>\n

                \n

                \ubb38\uc81c 3.10.<\/span>
                \n\\(I=\\{1,\\,2,\\,\\ldots,\\,n\\}\\)\uc774\uace0 \\(A\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc720\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

                  \n
                1. \\(n(A)\\), \\(n(I)\\), \\(n(A^I)\\) \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n
                2. \\(A^n\\), \\(A^I\\), \\(\\displaystyle\\prod_{i\\in I}A\\) \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\uace0, \\(A^n\\)\uacfc \\(A^I\\) \uc0ac\uc774\uc758 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c \ub300\uc751\uc744 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n

                  8. \ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uc131\uc9c8<\/h3>\n
                  \n

                  \uc815\ub9ac 3.2. (\ub370\uce74\ub974\ud2b8 \uacf1\uc758 \uc131\uc9c8)<\/span><\/p>\n

                  \uc9d1\ud569 \\(A,\\,B,\\,C,\\,D\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

                    \n
                  1. \\(A\\times(B\\cup C)=(A\\times B)\\cup(A\\times C)\\).<\/li>\n
                  2. \\(A\\times(B\\cap C)=(A\\times B)\\cap(A\\times C)\\).<\/li>\n
                  3. \\((A\\cup B)\\times C=(A\\times C)\\cup(B\\times C)\\).<\/li>\n
                  4. \\((A\\cap B)\\times C=(A\\times C)\\cap(B\\times C)\\).<\/li>\n
                  5. \\(A\\subseteq C\\)\uc774\uace0 \\(B\\subseteq D\\)\uc774\uba74 \\(A\\times B\\subseteq C\\times D\\)\uc774\ub2e4.<\/li>\n
                  6. \\(A\\times B=\\varnothing\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A=\\varnothing\\) \ub610\ub294 \\(B=\\varnothing\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
                    \n

                    \ubb38\uc81c 3.11.<\/span>
                    \n\uc704 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                    \ub370\uce74\ub974\ud2b8 \uacf1\uc5d0\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(A=\\{1\\}\\), \\(B=\\{2\\}\\)\uc774\uba74 \\(A\\times B=\\{(1,\\,2)\\}\\)\uc774\uace0 \\(B\\times A=\\{(2,\\,1)\\}\\)\uc774\ubbc0\ub85c \uc11c\ub85c \ub2e4\ub974\ub2e4. \ub2e4\ub9cc \\(A=B\\)\uc774\uac70\ub098 \\(A\\) \ub610\ub294 \\(B\\)\uac00 \uacf5\uc9d1\ud569\uc778 \uacbd\uc6b0\uc5d0\ub294 \ub450 \ub370\uce74\ub974\ud2b8 \uacf1\uc774 \uac19\ub2e4.<\/p>\n

                    \n

                    \ubb38\uc81c 3.12.<\/span>
                    \n\ucca8\uc790\uc9d1\ud569\uc774 \ub3d9\uc77c\ud55c \ub450 \uc9d1\ud569\uc871 \\(\\left\\{ A_i \\right\\}_{i\\in I}\\)\uc640 \\(\\left\\{ B_i \\right\\}_{i\\in I}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294\uc9c0 \ud655\uc778\ud558\uc2dc\uc624.
                    \n\\[\\left( \\prod_{i\\in I} A_i \\right) \\cap \\left( \\prod_{i\\in I} B_i \\right) = \\prod_{i\\in I} \\left( A_i \\cap B_i \\right)\\]
                    \n\uc131\ub9bd\ud55c\ub2e4\uba74 \uc99d\uba85\ud558\uace0, \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                    \n

                    \uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n

                    \uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span>
                    \n                    \uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span>
                    \n                    \ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span>
                    \n                    \uad00\uacc4\uc640 \ud568\uc218<\/a><\/span>
                    \n                    \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span>
                    \n                    \uc790\uc5f0\uc218<\/a><\/span>
                    \n                    \uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span>
                    \n                    \uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span>
                    \n                    \uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span>
                    \n                    \uc120\ud0dd \uacf5\ub9ac<\/a><\/span>
                    \n                    \ud615\uc2dd\ub17c\ub9ac<\/a><\/span>
                    \n                    \uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span>
                    \n                    \uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span>
                    \n                    \uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span>
                    \n                    \uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span>
                    \n                    \uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span>
                    \n                    \uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span>
                    \n                    \ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span>
                    \n                    \ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n

                    <\/p>\n","protected":false},"excerpt":{"rendered":"

                    \uc55e \uc7a5\uc5d0\uc11c\ub294 \ub450 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569 \ubc0f \uae30\ubcf8\uc801\uc778 \uc9d1\ud569 \uc5f0\uc0b0\uc758 \ubc95\uce59\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub97c \uc9d1\ud569\uc871\uc73c\ub85c \ud655\uc7a5\ud558\uc5ec \uc720\ud55c \uac1c \ub610\ub294 \uc784\uc758 \uac1c\uc758 \uc9d1\ud569\uc5d0 \ub300\ud55c \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc744 \uc815\uc758\ud558\uace0, \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774\uc5b4\uc11c \ub450 \uc9d1\ud569\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uacfc \uc77c\ubc18\ud654\ub41c \ub370\uce74\ub974\ud2b8 \uacf1\uc744 \ub2e4\ub8ec\ub2e4. 1. \uc720\ud55c \uac1c \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(A_1,\\, A_2,\\, \\ldots,\\, A_n\\)\uc774 \\(n\\)\uac1c\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \uc774\ub4e4 \uc9d1\ud569 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uc5d0 \uc18d\ud558\ub294…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":103,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9248","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9248","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9248"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9248\/revisions"}],"predecessor-version":[{"id":10078,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9248\/revisions\/10078"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9246"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}