{"id":8058,"date":"2021-09-30T14:43:53","date_gmt":"2021-09-30T05:43:53","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=8058"},"modified":"2022-01-12T16:17:01","modified_gmt":"2022-01-12T07:17:01","slug":"proof-of-inclusion-exclusion-principle","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/proof-of-inclusion-exclusion-principle\/","title":{"rendered":"\ud3ec\ud568-\ubc30\uc81c \uc6d0\ub9ac\uc758 \uc99d\uba85"},"content":{"rendered":"

\n\uc774 \uae00\uc5d0\uc11c\ub294 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc77c\ubc18\uc801\uc774\uace0 \uba85\ud655\ud558\uac8c \uae30\uc220\ud574\ubcf4\uace0 \ud3ec\ud568-\ubc30\uc81c \uc6d0\ub9ac\uc758 \uc0b0\ub73b\ud55c \uc99d\uba85\uc744 \uc2dc\ub3c4\ud574 \ubcf8\ub2e4. \uc774 \uae00\uc5d0\uc11c \uc18c\uac1c\ud55c \uc0b0\ub73b\ud55c \uc99d\uba85\uc740 \ucc38\uace0\ubb38\ud5cc [1]\uc744 \ubc14\ud0d5\uc73c\ub85c \uc791\uc131\ud55c \uac83\uc774\ub2e4.\n<\/p>\n

\ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc0b0\ub73b\ud558\uac8c \uae30\uc220\ud558\uae30<\/h4>\n

\n\ud559\uad50\uc218\ud559\uc5d0\uc11c\ub294 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc18c\uac1c\ud558\uace4 \ud55c\ub2e4.<\/span>1<\/sup><\/a><\/span>\n<\/p>\n

\n

\n \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac(\uc911,\uace0\ub4f1\ud559\uad50 \ubc84\uc804)<\/span>
\n\uc720\ud55c\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \uc9d1\ud569 \\(A, B\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[ \\vert A\\cup B\\vert=\\vert A\\vert +\\vert B\\vert – \\vert A\\cap B\\vert\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \uc720\ud55c\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \uc9d1\ud569 \\(A, B, C\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\vert A\\cup B\\cup C\\vert=\\vert A\\vert+\\vert B\\vert+\\vert C\\vert-\\vert A\\cap B\\vert-\\vert B\\cap C\\vert-\\vert C\\cap A\\vert+\\vert A\\cap B\\cap C\\vert \\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

\n\ucc38\uace0\ubb38\ud5cc [2]\uc5d0\uc11c\ub294 \ub354 \ub9ce\uc740 \uc9d1\ud569\uc758 \uac1c\uc218\ub97c \uc5fc\ub450\uc5d0 \ub454 \ub2e4\uc74c\uacfc \uac19\uc740 \ud615\ud0dc\ub85c \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc18c\uac1c\ud558\uace0 \uc788\ub2e4.\n<\/p>\n

\n

\n \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac(1). <\/span>
\n\uc720\ud55c\uc9d1\ud569\ub4e4 \\(A, B, C, \\ldots, Z\\) \uc911\uc5d0\uc11c \ud558\ub098 \uc774\uc0c1\uc758 \uc9d1\ud569\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c\uc758 \ucd1d \uac1c\uc218\ub294
\n\\begin{align*}
\n& +(\\vert A\\vert+\\vert B\\vert+\\vert C\\vert+\\cdots+\\vert Z\\vert) \\\\
\n& -(\\vert A\\cap B\\vert+\\vert A\\cap C\\vert+\\cdots+\\vert Y\\cap Z\\vert) \\\\
\n& +(\\vert A\\cap B\\cap C\\vert+\\vert A\\cap B\\cap D\\vert+\\cdots+\\vert X\\cap Y\\cap Z\\vert) \\\\
\n& -(\\vert A\\cap B\\cap C\\cap D\\vert+\\cdots+\\vert W\\cap X\\cap Y\\cap Z\\vert) \\\\
\n&\\ \\ \\vdots \\\\
\n& \\pm \\vert A\\cap B\\cap C\\cap \\cdots \\cap Y\\cap Z\\vert
\n\\end{align*}
\n\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n\uc774 \uacbd\uc6b0 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \ud559\uad50 \uc218\ud559 \ubc84\uc804\ubcf4\ub2e4\uc57c \ud6e8\uc52c \uc77c\ubc18\ud654\ub41c \ud615\ud0dc\ub85c \uae30\uc220\ud558\uae34 \ud558\uc600\uc9c0\ub9cc, \\(26\\)\uac1c\uc758 \uc54c\ud30c\ubcb3\uc744 \uc0ac\uc6a9\ud55c (\uac83\ucc98\ub7fc \ubcf4\uc778\ub2e4\ub294) \uc810 \ud639\uc740 \ub9e8 \ub9c8\uc9c0\ub9c9 \uc904\uc758 \uae30\ud638 ‘\\(\\pm\\)’\uc5d0\uc11c \ubaa8\uc885\uc758 \ubd88\ud3b8\ud568\uc744 \ub290\ub080 \ub3c5\uc790\ub3c4 \uc788\uc744 \uac83\uc774\ub2e4. \ub610\ud55c \uc704\uc758 \uae30\uc220\uc5d0\ub294 \uc5ec\ub7ec \uc904\uc744 \ud560\uc560\ud558\uc5ec \uacf5\uac04 \ubc0f \uc789\ud06c\ub97c \ub0ad\ube44\ud588\ub2e4\ub294 \uc544\uc26c\uc6b4 \uc810\ub3c4 \uc788\ub2e4.\n<\/p>\n

\n\ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\uc5d0 \ub4f1\uc7a5\ud558\ub294 \uc9d1\ud569\uc758 \uac1c\uc218\ub97c \\(n\\)\uc73c\ub85c \ub450\uc5b4 \uc55e\uc120 \ub450 \uacbd\uc6b0\ubcf4\ub2e4 \uc77c\ubc18\uc801\uc774\uace0 \uba85\ud655\ud558\uac8c \ub2e4\uc74c\uacfc \uac19\uc774 \uadf8 \uc6d0\ub9ac\ub97c \uae30\uc220\ud560 \uc218 \uc788\ub2e4.\n<\/p>\n

\n

\n \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac(2). <\/span>
\n\uc720\ud55c\uac1c\uc758 \uc720\ud55c \uc9d1\ud569 \\(A_{1}, A_{2}, \\ldots, A_{n}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\begin{multline*}
\n\\left\\vert\\bigcup_{j=1}^{n}A_{j} \\right\\vert=\\sum_{i=1}^{n}\\left\\vert A_{i}\\right\\vert-\\sum_{1\\leq i_{1}< i_{2}\\leq n}\\left\\vert A_{i_{1}}\\cap A_{i_{2}}\\right\\vert +\\sum_{1\\leq i_{1}< i_{2}< i_{3}\\leq n}\\left\\vert A_{i_{1}}\\cap A_{i_{2}}\\cap A_{i_{3}}\\right\\vert\\\\\n -\\cdots +(-1)^{n-1}\\left\\vert A_{1}\\cap A_{2}\\cap\\cdots\\cap A_{n}\\right\\vert\n\\end{multline*}\n<\/p>\n<\/div>\n

\n\uc704\uc758 \ub4f1\uc2dd\uc744 \uc880 \ub354 \uac04\ub7b5\ud558\uace0 \uba85\ud655\ud558\uac8c
\n\\[
\n\\left\\vert\\bigcup_{j=1}^{n}A_{j} \\right\\vert= \\sum_{j=1}^{n}(-1)^{j-1}\\sum_{1\\leq i_{1}<\\cdots < i_{j}\\leq n} \\left\\vert A_{i_{1}}\\cap \\cdots\\cap A_{i_{j}}\\right\\vert\n\\]\n\uc640 \uac19\uc774 \uc4f8 \uc218\ub3c4 \uc788\uaca0\ub2e4. \ubb3c\ub860 \uc5ec\uae30\uc11c \uae30\ud638\n\\[\n\\sum_{1\\leq i_{1}<\\cdots < i_{j}\\leq n}\n\\]\n\ub294 \\(1\\leq i_{1}<\\cdots < i_{j}\\leq n\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc591\uc758 \uc815\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \\(j\\)-tuple \\((i_{1},\\ldots,i_{j})\\)\uc5d0 \ub300\ud55c \ud569\uc73c\ub85c \ucd1d \\(\\binom{n}{j}\\)\uac1c\uc758 \ud56d\uc5d0 \ub300\ud55c \ud569\uc744 \ub098\ud0c0\ub0b8\ub2e4.\n<\/p>\n

\n\uc9d1\ud569 \\(X\\)\uc640 \uc74c\uc774 \uc544\ub2cc \uc815\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec, \\(k\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ubaa8\ub450 \ubaa8\uc544 \ub193\uc740 \uc9d1\ud569\uc744 \\(\\binom{X}{k}\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \uc704\uc758 \ud3ec\ud568-\ubc30\uc81c \uc6d0\ub9ac\uc758 \ub4f1\uc2dd\uc744
\n\\[
\n\\left\\vert\\bigcup_{j=1}^{n}A_{j} \\right\\vert= \\sum_{k=1}^{n}(-1)^{k-1}\\sum_{I\\in\\binom{\\{1,2,\\ldots,n\\}}{k}}\\left\\vert \\bigcap_{i\\in I}A_{i} \\right\\vert
\n\\]
\n\uc640 \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4. \uc774\uc815\ub3c4\uba74 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc5b4\ub290\uc815\ub3c4 \uc0b0\ub73b\ud558\uac8c \uae30\uc220\ud558\uc600\ub2e4 \ud560 \uc218 \uc788\uaca0\ub2e4. \ub9c8\uc9c0\ub9c9\uc73c\ub85c \ud569\uc758 \uae30\ud638 \\(\\sum\\)\ub97c \ud55c \ubc88\ub9cc \uc0ac\uc6a9\ud55c \uc0b0\ub73b\ud55c \ub4f1\uc2dd\uc744 \uac10\uc0c1\ud574\ubcf4\uc790.\n<\/p>\n

\n

\n \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac(\uc0b0\ub73b\ud55c \ubc84\uc804). <\/span>
\n\uc720\ud55c\uac1c\uc758 \uc720\ud55c \uc9d1\ud569 \\(A_{1}, A_{2}, \\ldots, A_{n}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[
\n\\left\\vert\\bigcup_{j=1}^{n}A_{j} \\right\\vert=\\sum_{\\varnothing\\neq I\\subset\\{1,2,\\ldots,n\\}}(-1)^{\\vert I\\vert-1}\\left\\vert \\bigcap_{i\\in I}A_{i}\\right\\vert
\n\\]\n<\/p>\n<\/div>\n

\n\uc704\uc758 \ub4f1\uc2dd\uc774 \uc790\uc5f0\uc2a4\ub7fd\ub2e4\uace0 \ub290\uaef4\uc9c8\ub54c\uae4c\uc9c0 \uc774\ub97c \uac10\uc0c1\ud558\ub294 \uac83\uc744 \ucd94\ucc9c\ud55c\ub2e4. \\(n=2\\) \ud639\uc740 \\(n=3\\)\uc778 \uacbd\uc6b0\ub97c \uc9c1\uc811 \uc368\ubcf4\uba74 \ub3c4\uc6c0\uc774 \ub41c\ub2e4.\n<\/p>\n

\ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc0b0\ub73b\ud558\uac8c \uc99d\uba85\ud558\uae30 <\/h4>\n

\n\ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub294 \ubb3c\ub860 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ubcf4\uc77c \uc218 \uc788\ub2e4. \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud55c \uc99d\uba85\uc740 \uac01\uc790 \uc2dc\ub3c4\ud574 \ubcf4\uae30 \ubc14\ub780\ub2e4(\ube14\ub85c\uadf8 \uae00\uc5d0\uc11c \uc660 \uacfc\uc81c?!). \ud55c\ud3b8 ‘\uc148\ud558\uae30\ub97c \uc774\uc6a9\ud55c \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac \uc99d\uba85’\ub3c4 \uc5bc\ub9c8\ub4e0\uc9c0 \ud560 \uc218 \uc788\ub2e4.\n<\/p>\n

\n\uc774 \uae00\uc5d0\uc11c \uc18c\uac1c\ud558\uace0\uc790 \ud558\ub294 \uc99d\uba85\uc740 \uc704\uc5d0\uc11c \uc5b8\uae09\ud55c \uc5b4\ub290 \uc99d\uba85\ubcf4\ub2e4\ub3c4 \ub354 \uc0b0\ub73b\ud55c \uae30\ubd84\uc744 \ub290\ub084 \uc218 \uc788\ub294 \uc99d\uba85\uc774\ub2e4. \uba3c\uc800 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ud655\uc778\ud558\uc790.
\n\\begin{equation}\\label{eq:identity01}
\n(1+x_{1})(1+x_{2})\\cdots(1+x_{n})=\\sum_{I\\subset\\{1,2,\\ldots,n\\}}\\left(\\prod_{i\\in I}x_{i}\\right) \\tag{1}
\n\\end{equation}
\n\uc5ec\uae30\uc11c \\(I=\\varnothing\\)\uc778 \uacbd\uc6b0 \\(\\prod_{i\\in \\varnothing}x_{i}=1\\)\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \uc774\ubc88\uc5d0\ub3c4 \uc704\uc758 \uc2dd \\eqref{eq:identity01}\uc774 \uc790\uc5f0\uc2a4\ub7fd\ub2e4\uace0 \ub290\ub084 \uc218 \uc788\ub3c4\ub85d \\(n=2\\) \ud639\uc740 \\(n=3\\)\uc778 \uacbd\uc6b0\ub97c \uc9c1\uc811 \uc368\ubcf4\ub294 \uac83\uc744 \ucd94\ucc9c\ud55c\ub2e4.\n<\/p>\n

\n

\uc99d\uba85(Proof of the PIE) <\/p>\n

\n   \uc9d1\ud569 \\(A\\)\ub97c \\(A=\\bigcup_{i=1}^{n}A_{i}\\)\ub85c \uc815\uc758\ud558\uace0 \ud568\uc218 \\(f_{i}:A\\to \\{0, 1\\}\\)\ub97c \uc9d1\ud569 \\(A_{i}\\)\uc758 \ud2b9\uc131\ud568\uc218, \uc989
\n\\[
\nf_{i}(a)=\\begin{cases}
\n1, & \\mbox{if \\(a\\in A_{i}\\)},\\\\
\n0, & \\mbox{otherwise}
\n\\end{cases}
\n\\]
\n\ub85c \uc815\uc758\ud558\uc790. \ubaa8\ub4e0 \\(a\\in A\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\prod_{i=1}^{n}\\left(1-f_{i}(a)\\right)=0
\n\\]
\n\uac00 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud560 \uc218 \uc788\uc73c\uba70, \ub610\ud55c \uc2dd \\eqref{eq:identity01}\uc5d0\uc11c \\(x_{i}=-f_{i}(a)\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc5ec
\n\\[
\n\\sum_{I\\subset\\{1,2,\\ldots,n\\}}(-1)^{\\vert I\\vert}\\prod_{i\\in I}f_{i}(a)=0
\n\\]
\n\uc744 \uc5bb\ub294\ub2e4. \uc704\uc758 \ub4f1\uc2dd\uc744 \ubaa8\ub4e0 \\(a\\)\uc5d0 \ub300\ud558\uc5ec \ud569\ud55c \ud6c4, \ud569\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4
\n\\begin{equation}\\label{eq:identity02}
\n\\begin{split}
\n0&=\\sum_{a\\in A}\\left(\\sum_{I\\subset\\{1,2,\\ldots,n\\}}(-1)^{\\vert I\\vert}\\prod_{i\\in I}f_{i}(a)\\right) \\\\
\n &=\\sum_{I\\subset\\{1,2,\\ldots,n\\}}(-1)^{\\vert I\\vert}\\left(\\sum_{a\\in A}\\prod_{i\\in I}f_{i}(a)\\right)
\n\\end{split} \\tag{2}
\n\\end{equation}
\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

\n\uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c \\(a\\)\ub97c \\(\\prod_{i\\in I}f_{i}(a)\\)\ub85c \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \\(\\bigcap_{i\\in I}A_{i}\\)\uc758 \ud2b9\uc131\ud568\uc218\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\uace0 \ub530\ub77c\uc11c \\[ \\sum_{a\\in A}\\prod_{i\\in I}f_{i}(a)=\\left\\vert\\bigcap_{i\\in I}A_{i}\\right\\vert\\]\ub97c \uc5bb\ub294\ub2e4. \ub610\ud55c \uc774\ub54c \\(I=\\varnothing\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc5ec
\n\\[
\n\\sum_{a\\in A}\\prod_{i\\in I}f_{i}(a)=\\sum_{a\\in A}1=\\left\\vert A\\right\\vert
\n\\]
\n\ub3c4 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \uc2dd \\eqref{eq:identity02}\ub97c
\n\\[
\n\\vert A\\vert +\\sum_{\\varnothing\\neq I\\subset\\{1,2,\\ldots,n\\}}(-1)^{\\vert I\\vert}\\left\\vert\\bigcap_{i\\in I}A_{i}\\right\\vert=0
\n\\]
\n\ub85c \ub2e4\uc2dc \uc4f8 \uc218 \uc788\uc73c\uba70 \uc774\ub294 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \ub098\ud0c0\ub0b4\ub294 \ub4f1\uc2dd\uacfc \ub3d9\uc77c\ud55c \uc2dd\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\ucc38\uace0 \ubb38\ud5cc <\/h3>\n
    \n
  1. Matousek, J., & Nesetril, J. (2009). Invitation to discrete mathematics. Oxford University Press. <\/li>\n
  2. Martin, G. E. (2001). Counting: The art of enumerative combinatorics. Springer Science & Business Media. <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    \uc774 \uae00\uc5d0\uc11c\ub294 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc77c\ubc18\uc801\uc774\uace0 \uba85\ud655\ud558\uac8c \uae30\uc220\ud574\ubcf4\uace0 \ud3ec\ud568-\ubc30\uc81c \uc6d0\ub9ac\uc758 \uc0b0\ub73b\ud55c \uc99d\uba85\uc744 \uc2dc\ub3c4\ud574 \ubcf8\ub2e4. \uc774 \uae00\uc5d0\uc11c \uc18c\uac1c\ud55c \uc0b0\ub73b\ud55c \uc99d\uba85\uc740 \ucc38\uace0\ubb38\ud5cc [1]\uc744 \ubc14\ud0d5\uc73c\ub85c \uc791\uc131\ud55c \uac83\uc774\ub2e4. \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \uc0b0\ub73b\ud558\uac8c \uae30\uc220\ud558\uae30 \ud559\uad50\uc218\ud559\uc5d0\uc11c\ub294 \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc18c\uac1c\ud558\uace4 \ud55c\ub2e4. \ud3ec\ud568-\ubc30\uc81c\uc758 \uc6d0\ub9ac(\uc911,\uace0\ub4f1\ud559\uad50 \ubc84\uc804) \uc720\ud55c\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \uc9d1\ud569 \\(A, B\\)\uc5d0 \ub300\ud558\uc5ec \\( \\vert A\\cup B\\vert=\\vert A\\vert +\\vert B\\vert – \\vert A\\cap B\\vert\\) \uac00 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \uc720\ud55c\uac1c\uc758 \uc6d0\uc18c\ub97c \uac16\ub294 \uc9d1\ud569 \\(A, B,…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[54],"tags":[563,558,560,562,561,559],"class_list":["post-8058","post","type-post","status-publish","format-standard","hentry","category-basic-mathematics","tag-counting","tag-pie","tag-the-principle-of-inclusion-and-exclusion","tag-562","tag-561","tag-559"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8058","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=8058"}],"version-history":[{"count":1,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8058\/revisions"}],"predecessor-version":[{"id":8059,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8058\/revisions\/8059"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}