{"id":9244,"date":"2025-10-17T19:34:05","date_gmt":"2025-10-17T10:34:05","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9244"},"modified":"2026-09-27T11:43:39","modified_gmt":"2026-09-27T02:43:39","slug":"ch02-sets","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch02-sets\/","title":{"rendered":"\uc9d1\ud569\uc758 \uac1c\ub150"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>2. \uc9d1\ud569\uc758 \uac1c\ub150<\/h2>\n\n --><\/p>\n<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uba85\uc81c\uc640 \ub17c\ub9ac \uc5f0\uc0b0, \uba85\uc81c \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774\uc81c \uc774\ub7ec\ud55c \ub17c\ub9ac\uc758 \uc5b8\uc5b4\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc218\ud559\uc758 \uae30\ubcf8 \ub300\uc0c1\uc778 \uc9d1\ud569\uc744 \ub2e4\ub8ec\ub2e4. 19\uc138\uae30 \ub9d0 \uce78\ud1a0\uc5b4(Georg Cantor)\uac00 \ubc1c\uc804\uc2dc\ud0a8 \uc9d1\ud569\ub860\uc740 \ud604\ub300 \uc218\ud559\uc758 \uac70\uc758 \ubaa8\ub4e0 \ubd84\uc57c\uc758 \uae30\ubc18\uc774 \ub418\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uc5f0\uc0b0\uc744 \ub2e4\ub8e8\uba70, \uba85\uc81c\uc640 \uc9d1\ud569 \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>1. \uc9d1\ud569\uacfc \uc6d0\uc18c<\/h3>\n<p>\uc774 \uc7a5\uc5d0\uc11c\ub294 <span class=\"defined\">\uc9d1\ud569<\/span>(set)\uc744 \uba85\ud655\ud558\uac8c \uad6c\ubcc4\ub418\ub294 \ub300\uc0c1\ub4e4\uc758 \ubaa8\uc784\uc73c\ub85c \uc9c1\uad00\uc801\uc73c\ub85c \uc774\ud574\ud55c\ub2e4. \uc9d1\ud569\uc744 \uc774\ub8e8\ub294 \uac01\uac01\uc758 \ub300\uc0c1\uc744 <span class=\"defined\">\uc6d0\uc18c<\/span>(element)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9d1\ud569\uc740 \ubcf4\ud1b5 \ub300\ubb38\uc790 \\(A\\), \\(B\\), \\(C\\) \ub4f1\uc73c\ub85c \ub098\ud0c0\ub0b4\uace0, \uc6d0\uc18c\ub294 \uc18c\ubb38\uc790 \\(a\\), \\(b\\), \\(c\\) \ub4f1\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub7ec\ud55c \uc9c1\uad00\uc801 \uc124\uba85\uc744 \uacf5\ub9ac\uc801\uc73c\ub85c \uc815\uad50\ud558\uac8c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc740 \ub4a4\uc758 \uacf5\ub9ac\uc801 \uc9d1\ud569\ub860\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.<\/p>\n<p>\uc6d0\uc18c \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud560 \ub54c \\(a \\in A\\)\ub85c \ub098\ud0c0\ub0b4\uace0, \uc18d\ud558\uc9c0 \uc54a\uc744 \ub54c \\(a \\notin A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(A\\)\uac00 \uc138 \uac1c\uc758 \uc790\uc5f0\uc218 \\(1,\\) \\(2,\\) \\(3\\)\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc77c \ub54c \\(2 \\in A\\)\uc774\uace0 \\(4 \\notin A\\)\uc774\ub2e4.<\/p>\n<p>\uc6d0\uc18c\ub97c \ud558\ub098\ub3c4 \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc744 <span class=\"defined\">\uacf5\uc9d1\ud569<\/span>(empty set)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\varnothing\\) \ub610\ub294 \\(\\{\\}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul style=\"list-style-type: disc; margin-top: 0.5em;\">\n<li style=\"margin-bottom: 0.5em;\"><span class=\"defined\">\uc6d0\uc18c\ub098\uc5f4\ubc95<\/span>: \uc9d1\ud569\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \ub098\uc5f4\ud558\uace0 \uc911\uad04\ud638\ub85c \ubb36\uc5b4 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\uc608\ub97c \ub4e4\uc5b4 \\(5\\) \uc774\ud558\uc778 \uc591\uc758 \uc815\uc218\uc758 \ubaa8\uc784\uc740 \\(\\{1,\\, 2,\\, 3,\\, 4,\\, 5\\}\\)\uc774\ub2e4.<\/li>\n<li style=\"margin-bottom: 0.5em;\"><span class=\"defined\">\uc870\uac74\uc81c\uc2dc\ubc95<\/span>: \uc6d0\uc18c\uac00 \ub9cc\uc871\uc2dc\ucf1c\uc57c \ud560 \uc870\uac74\uc744 \uc81c\uc2dc\ud55c\ub2e4.<br \/>\n\uc608\ub97c \ub4e4\uc5b4 \\(5\\) \uc774\ud558\uc778 \uc591\uc758 \uc815\uc218\uc758 \ubaa8\uc784\uc740 \\(\\{x \\mid x\\text{\ub294 } 5 \\text{ \uc774\ud558\uc778 \uc591\uc758 \uc815\uc218}\\}\\)\uc774\ub2e4.<br \/>\n[\uc6d0\ub798\ub294 \\(\\left\\{ x\\in\\mathbb{Z} \\mid 0 < x \\le 5\\right\\}\\)\uc640 \uac19\uc774 \\(x\\)\uac00 \uc18d\ud558\ub294 \uc9d1\ud569\uc744 \uba85\uc2dc\ud574 \uc8fc\uc5b4\uc57c \ud55c\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\/\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a> \uc7a5\uc758 \ubd84\ub9ac \uacf5\ub9ac\uaf34\uc744 \ucc38\uc870\ud558\uae30 \ubc14\ub780\ub2e4.]<\/li>\n<\/ul>\n<p>\uc55e \uc7a5\uc5d0\uc11c \uc815\uc758\ud55c \uba85\uc81c \ud568\uc218 \\(p(x)\\)\uc640 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec, &#8220;\\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(p(x)\\)\uac00 \ucc38\uc774\ub2e4&#8221;\ub97c \\(\\forall x\\in A,\\,p(x)\\)\ub85c \ub098\ud0c0\ub0b4\uace0, &#8220;\\(p(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(A\\)\uc758 \uc6d0\uc18c \\(x\\)\uac00 \uc801\uc5b4\ub3c4 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4&#8221;\ub97c \\(\\exists x\\in A,\\,p(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ubc94\uc704\uac00 \ubb38\ub9e5\uc0c1 \uc815\ud574\uc838 \uc788\uc73c\uba74 \uac01\uac01 \\(\\forall x\\,p(x)\\), \\(\\exists x\\,p(x)\\)\uc640 \uac19\uc774 \uc4f0\uae30\ub3c4 \ud55c\ub2e4. \uc774 \uae30\ud638\ub4e4\uc740 \uc774 \uc7a5\uc758 \ub4a4\uc5d0\uc11c \ud55c\uc815\uba85\uc81c\ub97c \ub2e4\ub8f0 \ub54c \uc790\uc138\ud788 \uc124\uba85\ud55c\ub2e4.<\/p>\n<h3>2. \uc9d1\ud569 \uc0ac\uc774\uc758 \uad00\uacc4<\/h3>\n<p>\uc9d1\ud569 \\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uc9d1\ud569 \\(B\\)\uc758 \uc6d0\uc18c\uc77c \ub54c, \\(A\\)\ub97c \\(B\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc9d1\ud569<\/span>(subset)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A \\subseteq B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\uc77c\ubd80 \ubb38\ud5cc\uc5d0\uc11c\ub294 \\(A\\subset B\\)\ub97c \\(A\\subseteq B\\)\uc640 \uac19\uc740 \ub73b\uc73c\ub85c \uc0ac\uc6a9\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc774 \ucc45\uc5d0\uc11c\ub294 \\(A\\subset B\\)\ub97c \uc9c4\ubd80\ubd84\uc9d1\ud569\uc758 \ub73b\uc73c\ub85c \uc0ac\uc6a9\ud55c\ub2e4.] \ubd80\ubd84\uc9d1\ud569\uc758 \uc815\uc758\ub97c \uae30\ud638\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[A \\subseteq B \\quad \\Longleftrightarrow \\quad \\forall x\\in A,\\, x\\in B\\]<\/p>\n<p>\\(A \\subseteq B\\)\uc774\uba74\uc11c \\(A \\neq B\\)\uc77c \ub54c \\(A\\)\ub97c \\(B\\)\uc758 <span class=\"defined\">\uc9c4\ubd80\ubd84\uc9d1\ud569<\/span>(proper subset)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A \\subsetneq B\\) \ub610\ub294 \\(A \\subset B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uac00 <span class=\"defined\">\uac19\ub2e4<\/span>\ub294 \uac83\uc740 \\(A \\subseteq B\\)\uc774\uace0 \\(B \\subseteq A\\)\uc778 \uac83\uc774\ub2e4. \ub450 \uc9d1\ud569\uc774 \uac19\ub2e4\ub294 \uc815\uc758\ub97c \uae30\ud638\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[A = B \\quad \\Longleftrightarrow \\quad A \\subseteq B \\,\\,\\,\\wedge\\,\\,\\, B \\subseteq A\\]<br \/>\n\uc774\uac83\uc740 \ub610\ud55c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4.<br \/>\n\\[A = B \\quad \\Longleftrightarrow \\quad \\forall x\\, (x\\in A \\,\\, \\leftrightarrow x\\in B)\\]<br \/>\n\uc704 \ub450 \uc2dd \uc911 \ud558\ub098\ub97c \uc815\uc758\ub85c \uc0bc\uc73c\uba74 \ub2e4\ub978 \ud558\ub098\ub294 \uc815\ub9ac\uac00 \ub41c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.1.<\/span><br \/>\n\uc9d1\ud569 \\(A=\\{1,\\,2,\\,\\{1,2\\}\\}\\)\ub77c \ud558\uc790. \ub2e4\uc74c \uba85\uc81c\uac00 \ucc38\uc778\uc9c0 \uac70\uc9d3\uc778\uc9c0 \ud310\ub2e8\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(1\\in A\\)<\/li>\n<li>\\(\\{1\\}\\in A\\)<\/li>\n<li>\\(\\{1\\}\\subseteq A\\)<\/li>\n<li>\\(\\{1,2\\}\\in A\\)<\/li>\n<li>\\(\\{1,2\\}\\subseteq A\\)<\/li>\n<li>\\(\\varnothing\\in A\\)<\/li>\n<li>\\(\\varnothing\\subseteq A\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.2.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(\\varnothing \\subseteq A\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(A \\subseteq A\\)\uc774\ub2e4.<\/li>\n<li>\\(A \\subseteq B\\)\uc774\uace0 \\(B \\subseteq C\\)\uc774\uba74 \\(A \\subseteq C\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>3. \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/h3>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc9d1\ud569\uc758 \uc5f0\uc0b0\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul style=\"list-style-type: disc; margin-top: 0.5em;\">\n<li style=\"margin-bottom: 0.5em;\">\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\ud569\uc9d1\ud569<\/span>(union) \\(A \\cup B\\)\ub294 \\(A\\)\ub098 \\(B\\) \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<br \/>\n\\[A \\cup B = \\{x \\mid x \\in A \\;\\text{ \ub610\ub294 }\\; x \\in B\\}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\">\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\uad50\uc9d1\ud569<\/span>(intersection) \\(A \\cap B\\)\ub294 \\(A\\)\uc640 \\(B\\) \ubaa8\ub450\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<br \/>\n\\[A \\cap B = \\{x \\mid x \\in A \\;\\text{ \uadf8\ub9ac\uace0 }\\; x \\in B\\}\\]<br \/>\n\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(A \\cap B = \\varnothing\\)\uc77c \ub54c, \\(A\\)\uc640 \\(B\\)\ub294 <span class=\"defined\">\uc11c\ub85c\uc18c<\/span>(disjoint)\ub77c\uace0 \ud55c\ub2e4.\n<\/li>\n<li style=\"margin-bottom: 0.5em;\">\uc9d1\ud569 \\(A\\)\uc5d0\uc11c \uc9d1\ud569 \\(B\\)\ub97c \ube80 <span class=\"defined\">\ucc28\uc9d1\ud569<\/span>(difference) \\(A \\setminus B\\) \ub610\ub294 \\(A &#8211; B\\)\ub294 \\(A\\)\uc5d0\ub294 \uc18d\ud558\uc9c0\ub9cc \\(B\\)\uc5d0\ub294 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<br \/>\n\\[A \\setminus B = \\{x \\mid x \\in A \\;\\text{ \uadf8\ub9ac\uace0 }\\; x \\notin B\\}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\">\uc804\uccb4\uc9d1\ud569 \\(U\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc9d1\ud569 \\(A\\)\uc758 <span class=\"defined\">\uc5ec\uc9d1\ud569<\/span>(complement) \\(A^c\\) \ub610\ub294 \\(\\overline{A}\\)\ub294 \\(U\\)\uc5d0\ub294 \uc18d\ud558\uc9c0\ub9cc \\(A\\)\uc5d0\ub294 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<br \/>\n\\[A^c = U \\setminus A = \\{x \\in U \\mid x \\notin A\\}\\]\n<\/li>\n<\/ul>\n<p>\uc9d1\ud569\uc758 \uae30\ubcf8 \uc5f0\uc0b0 \ubc95\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 2.1. (\uc9d1\ud569 \uc5f0\uc0b0\uc758 \ubc95\uce59)<\/span><\/p>\n<p>\uc804\uccb4\uc9d1\ud569 \\(U\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \\(A,B,C\\subseteq U\\)\ub77c \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li><span class=\"defined\">\uad50\ud658\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup B &#038;= B \\cup A\\\\[3pt]<br \/>\nA \\cap B &#038;= B \\cap A<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\uacb0\ud569\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\n(A \\cup B) \\cup C &#038;= A \\cup (B \\cup C)\\\\[3pt]<br \/>\n(A \\cap B) \\cap C &#038;= A \\cap (B \\cap C)<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\ubd84\ubc30\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup (B \\cap C) &#038;= (A \\cup B) \\cap (A \\cup C)\\\\[3pt]<br \/>\nA \\cap (B \\cup C) &#038;= (A \\cap B) \\cup (A \\cap C)<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\ub4dc \ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\n(A \\cup B)^c &#038;= A^c \\cap B^c\\\\[3pt]<br \/>\n(A \\cap B)^c &#038;= A^c \\cup B^c<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\ud56d\ub4f1\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup \\varnothing &#038;= A\\\\[3pt]<br \/>\nA \\cap U &#038;= A<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\uc9c0\ubc30\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup U &#038;= U\\\\[3pt]<br \/>\nA \\cap \\varnothing &#038;= \\varnothing<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\uba71\ub4f1\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup A &#038;= A\\\\[3pt]<br \/>\nA \\cap A &#038;= A<br \/>\n\\end{aligned}\\]\n<\/li>\n<li><span class=\"defined\">\uc5ec\uc9d1\ud569\ubc95\uce59<\/span>:<br \/>\n\\[\\begin{aligned}<br \/>\nA \\cup A^c &#038;= U\\\\[3pt]<br \/>\nA \\cap A^c &#038;= \\varnothing\\\\[3pt]<br \/>\n(A^c)^c &#038;= A<br \/>\n\\end{aligned}\\]\n<\/li>\n<\/ol>\n<\/div>\n<p>\uc774\ub7ec\ud55c \uc9d1\ud569 \uc5f0\uc0b0\uc758 \ubc95\uce59\ub4e4\uc740 1\uc7a5\uc5d0\uc11c \ub2e4\ub8ec \uba85\uc81c \uc5f0\uc0b0\uc758 \ubc95\uce59\ub4e4\uacfc \uc77c\ub300\uc77c \ub300\uc751 \uad00\uacc4\uc5d0 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc9d1\ud569 \uc5f0\uc0b0\uc758 \ubd84\ubc30\ubc95\uce59 \uc911 \uccab \ubc88\uc9f8 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uba85\uc81c \uc5f0\uc0b0\uc758 \ubd84\ubc30\ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ub41c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A \\cup (B \\cap C ) &#038;\\,\\,\\Longleftrightarrow\\,\\, x\\in A \\,\\vee\\, x\\in B\\cap C &#038;&#038;\\text{(\ud569\uc9d1\ud569\uc758 \uc815\uc758)}\\\\[3pt]<br \/>\n&#038;\\,\\,\\Longleftrightarrow\\,\\, x\\in A \\,\\vee\\, (x\\in B \\,\\wedge\\, x\\in C) &#038;&#038;\\text{(\uad50\uc9d1\ud569\uc758 \uc815\uc758)}\\\\[3pt]<br \/>\n&#038;\\,\\,\\Longleftrightarrow\\,\\, (x\\in A \\,\\vee\\, x\\in B ) \\,\\wedge\\, (x\\in A \\,\\vee\\, x\\in C) &#038;&#038;\\text{(\uba85\uc81c \uc5f0\uc0b0\uc758 \ubd84\ubc30\ubc95\uce59)}\\\\[3pt]<br \/>\n&#038;\\,\\,\\Longleftrightarrow\\,\\, (x\\in A \\cup B ) \\,\\wedge\\, (x\\in A \\cup C) &#038;&#038;\\text{(\ud569\uc9d1\ud569\uc758 \uc815\uc758)} \\\\[3pt]<br \/>\n&#038;\\,\\,\\Longleftrightarrow\\,\\, x\\in (A \\cup B ) \\cap (A \\cup C) . &#038;&#038;\\text{(\uad50\uc9d1\ud569\uc758 \uc815\uc758)}<br \/>\n\\end{aligned}\\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.3.<\/span><br \/>\n\uc804\uccb4\uc9d1\ud569 \\(U=\\{1,\\,2,\\,\\ldots,\\,10\\}\\)\uacfc \uadf8 \ubd80\ubd84\uc9d1\ud569<br \/>\n\\[<br \/>\nA=\\{1,\\,2,\\,3,\\,4,\\,5\\},\\quad B=\\{4,\\,5,\\,6,\\,7\\},\\quad C=\\{2,\\,4,\\,6,\\,8,\\,10\\}<br \/>\n\\]<br \/>\n\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc9d1\ud569\uc744 \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A\\cup B\\), \\(A\\cap B\\)<\/li>\n<li>\\(A\\setminus B\\), \\(B\\setminus A\\)<\/li>\n<li>\\(A^c\\), \\(B^c\\)<\/li>\n<li>\\((A\\cup B)^c\\), \\(A^c\\cap B^c\\)<\/li>\n<li>\\((A\\cap C)\\setminus B\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.4.<\/span><br \/>\n\uba85\uc81c \uc5f0\uc0b0\uc758 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A \\cap (B \\cup C)=(A\\cap B)\\cup(A\\cap C)\\)<\/li>\n<li>\\((A\\cup B)^c=A^c\\cap B^c\\)<\/li>\n<li>\\(A\\cup A^c=U\\)<\/li>\n<li>\\((A^c)^c=A\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.5.<\/span><br \/>\n\uc9d1\ud569\uc758 \uc5f0\uc0b0\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((A \\cup B) \\setminus C = (A \\setminus C) \\cup (B \\setminus C)\\)<\/li>\n<li>\\((A \\setminus B) \\cap C = (A \\cap C) \\setminus B\\)<\/li>\n<li>\\(A \\setminus (B \\cup C) = (A \\setminus B) \\cap (A \\setminus C)\\)<\/li>\n<\/ol>\n<\/div>\n<h3>4. \uba71\uc9d1\ud569<\/h3>\n<p>\uc9d1\ud569 \\(A\\)\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \uc9d1\ud569\uc744 \\(A\\)\uc758 <span class=\"defined\">\uba71\uc9d1\ud569<\/span>(power set)\uc774\ub77c \ud558\uace0 \\(\\mathcal{P}(A)\\) \ub610\ub294 \\(2^A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(A\\)\uc758 \uba71\uc9d1\ud569\uc744 \uc870\uac74\uc81c\uc2dc\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\mathcal{P}(A) = \\{X \\mid X \\subseteq A\\}\\]<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4, \\(A = \\{1,\\, 2\\}\\)\uc77c \ub54c,<br \/>\n\\[\\mathcal{P}(A) = \\{\\varnothing,\\, \\{1\\},\\, \\{2\\},\\, \\{1,\\, 2\\}\\} .\\]<\/p>\n<p>\uba71\uc9d1\ud569\uc774 \uac00\uc9c4 \uc911\uc694\ud55c \uc131\uc9c8\ub85c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\\(\\varnothing \\in \\mathcal{P}(A)\\)\uc774\uace0 \\(A \\in \\mathcal{P}(A)\\)\uc774\ub2e4.<\/li>\n<li>\\(X \\in \\mathcal{P}(A)\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(X \\subseteq A\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(A \\subseteq B\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\mathcal{P}(A) \\subseteq \\mathcal{P}(B)\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.6.<\/span><br \/>\n\uc9d1\ud569 \\(A=\\{a,b,c\\}\\)\uc640 \\(B=\\{a,b\\}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \uad6c\ud558\uac70\ub098 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\mathcal{P}(A)\\)\ub97c \uc6d0\uc18c\ub098\uc5f4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n<li>\\(A\\)\uc758 \uc6d0\uc18c\ub97c \uc815\ud655\ud788 \ub450 \uac1c \uac00\uc9c0\ub294 \ubd80\ubd84\uc9d1\ud569\uc744 \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(\\mathcal{P}(B)\\)\ub97c \uad6c\ud558\uace0 \\(\\mathcal{P}(B)\\subseteq\\mathcal{P}(A)\\)\uc784\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<li>\uc784\uc758\uc758 \uc9d1\ud569 \\(X\\), \\(Y\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nX\\subseteq Y \\quad\\Longleftrightarrow\\quad \\mathcal{P}(X)\\subseteq\\mathcal{P}(Y)<br \/>\n\\]<br \/>\n\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<h3>5. \uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/h3>\n<p>\uc9d1\ud569\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c <span class=\"defined\">\uc720\ud55c\uc9d1\ud569<\/span>(finite set)\uacfc <span class=\"defined\">\ubb34\ud55c\uc9d1\ud569<\/span>(infinite set)\uc73c\ub85c \ubd84\ub958\ub41c\ub2e4. \uc6d0\uc18c\ub97c \ubaa8\ub450 \uc138\uc5c8\uc744 \ub54c \uadf8 \uac1c\uc218\uac00 \uc5b4\ub5a4 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\uac00 \ub418\ub294 \uc9d1\ud569\uc744 \uc720\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uadf8\ub807\uc9c0 \uc54a\uc740 \uc9d1\ud569\uc744 \ubb34\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc720\ud55c\uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \\(n(A)\\) \ub610\ub294 \\(|A|\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud2b9\ud788 \\(n(\\varnothing) = 0\\)\uc774\ub2e4.<\/p>\n<p>\uc720\ud55c\uc9d1\ud569 \\(A\\), \\(B\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(n(A \\cup B) = n(A) + n(B) &#8211; n(A \\cap B)\\).<\/li>\n<li>\\(A \\cap B = \\varnothing\\)\uc774\uba74 \\(n(A \\cup B) = n(A) + n(B)\\)\uc774\ub2e4.<\/li>\n<li>\uc720\ud55c\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(n(\\mathcal{P}(A)) = 2^{n(A)}\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ubb34\ud55c\uc9d1\ud569\uc758 \uc608\ub85c\ub294 \uc790\uc5f0\uc218\uc758 \uc9d1\ud569 \\(\\mathbb{N}\\), \uc815\uc218\uc758 \uc9d1\ud569 \\(\\mathbb{Z}\\), \uc720\ub9ac\uc218\uc758 \uc9d1\ud569 \\(\\mathbb{Q}\\), \uc2e4\uc218\uc758 \uc9d1\ud569 \\(\\mathbb{R}\\) \ub4f1\uc774 \uc788\ub2e4. [\uc774 \uad50\uc7ac\uc5d0\uc11c &#8216;\uc790\uc5f0\uc218&#8217;\ub294 \\(0\\) \uc774\uc0c1\uc778 \uc815\uc218\ub97c \ub098\ud0c0\ub0b4\uba70, \\(\\mathbb{N} = \\left\\{0,\\,1,\\,2,\\,3,\\,\\ldots\\right\\}\\)\uc774\ub2e4.] \ubb34\ud55c\uc9d1\ud569\ub3c4 \uadf8 \ud06c\uae30\uc5d0 \ub530\ub77c \uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\ubb34\ud55c\uc9d1\ud569\uc73c\ub85c \ubd84\ub958\ub418\ub294\ub370, \uc774\uac83\uc740 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a> \uc7a5\uc5d0\uc11c \uc790\uc138\ud788 \ub2e4\ub8ec\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.7.<\/span><br \/>\n\uc9d1\ud569 \\(S=\\{1,2,\\ldots,60\\}\\)\uc5d0\uc11c \\(2\\)\uc758 \ubc30\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \\(A\\), \\(3\\)\uc758 \ubc30\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \\(B\\)\ub77c \ud558\uc790. \ub2e4\uc74c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(n(A)\\)\uc640 \\(n(B)\\)<\/li>\n<li>\\(n(A\\cap B)\\)<\/li>\n<li>\\(n(A\\cup B)\\)<\/li>\n<li>\\(2\\)\uc758 \ubc30\uc218\ub3c4 \uc544\ub2c8\uace0 \\(3\\)\uc758 \ubc30\uc218\ub3c4 \uc544\ub2cc \\(S\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.8.<\/span><br \/>\n\uc720\ud55c\uc9d1\ud569 \\(A\\), \\(B\\), \\(C\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(n(A \\cup B \\cup C) = n(A) + n(B) + n(C) &#8211; n(A \\cap B) &#8211; n(B \\cap C) &#8211; n(C \\cap A) + n(A \\cap B \\cap C)\\)<\/li>\n<li>\\(A \\subseteq B\\)\uc774\uba74 \\(n(A) \\leq n(B)\\)\uc774\ub2e4.<\/li>\n<li>\ub450 \uc9d1\ud569 \\(A\\), \\(B\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc744 \\(A\\times B=\\{(a,b)\\mid a\\in A,\\ b\\in B\\}\\)\ub85c \uc815\uc758\ud560 \ub54c, \\(n(A \\times B) = n(A) \\cdot n(B)\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.9.<\/span><br \/>\n\uc591\uc758 \uc815\uc218 \\(m\\)\uacfc \uc720\ud55c\uc9d1\ud569 \\(A_1,\\,\\ldots,\\,A_m\\)\uc5d0 \ub300\ud55c &#8216;\ud3ec\ud568&#8211;\ubc30\uc81c\uc758 \uc6d0\ub9ac&#8217;\ub97c \uc11c\uc220\ud558\uace0, \\(m=3\\)\uc778 \uacbd\uc6b0\uc5d0 \ubb38\uc81c 2.8\uc758 (1)\uc774 \uc5bb\uc5b4\uc9d0\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>6. \ud55c\uc815\uba85\uc81c<\/h3>\n<p>\uc55e\uc5d0\uc11c \uc0ac\uc6a9\ud55c \\(\\forall\\)\uc640 \\(\\exists\\)\ub294 \uc9d1\ud569\uacfc \uad00\ub828\ud558\uc5ec \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 <span class=\"defined\">\ud55c\uc815\uae30\ud638<\/span>(quantifier)\uc774\ub2e4. \ud55c\uc815\uae30\ud638\ub97c \uc0ac\uc6a9\ud55c \uba85\uc81c\ub97c <span class=\"defined\">\ud55c\uc815\uba85\uc81c<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c\ub294 \uc804\uce6d\uba85\uc81c\uc640 \uc874\uc7ac\uba85\uc81c\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h4>\uc804\uce6d\uba85\uc81c<\/h4>\n<p>\uba85\uc81c \ud568\uc218 \\(p(x)\\)\uc640 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec, &#8220;\\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(p(x)\\)\uac00 \ucc38\uc774\ub2e4&#8221;\ub97c<br \/>\n\\[\\forall x \\in A,\\, p(x)\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud45c\uae30\ub294 \\(\\forall x\\,(x\\in A\\to p(x))\\)\uc758 \uc57d\uc2dd \ud45c\uae30\uc774\ub2e4. \uc774\ub54c \\(\\forall\\)\ub294 <span class=\"defined\">\uc804\uce6d\uae30\ud638<\/span>\ub77c\uace0 \ubd80\ub974\uba70, &#8220;\ubaa8\ub4e0&#8221;\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc804\uce6d\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud45c\ud604\ud55c \uba85\uc81c\ub97c <span class=\"defined\">\uc804\uce6d\uba85\uc81c<\/span>(universal statement)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ubaa8\ub4e0 \uc9dd\uc218\uc758 \uc9d1\ud569\uc744 \\(E\\)\ub77c\uace0 \ud560 \ub54c, &#8220;\ubaa8\ub4e0 \uc9dd\uc218\ub294 2\ub85c \ub098\ub204\uc5b4\ub5a8\uc5b4\uc9c4\ub2e4&#8221;\ub77c\ub294 \ubb38\uc7a5\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[\\forall n \\in E,\\,\\, 2 \\mid n \\]<\/p>\n<h4>\uc874\uc7ac\uba85\uc81c<\/h4>\n<p>\uba85\uc81c \ud568\uc218 \\(q(x)\\)\uc640 \uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec, &#8220;\\(q(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(A\\)\uc758 \uc6d0\uc18c \\(x\\)\uac00 \uc801\uc5b4\ub3c4 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4&#8221;\ub97c<br \/>\n\\[\\exists x \\in A,\\, q(x)\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud45c\uae30\ub294 \\(\\exists x\\,(x\\in A\\wedge q(x))\\)\uc758 \uc57d\uc2dd \ud45c\uae30\uc774\ub2e4. \uc774\ub54c \\(\\exists\\)\ub294 <span class=\"defined\">\uc874\uc7ac\uae30\ud638<\/span>\ub77c\uace0 \ubd80\ub974\uba70, &#8220;\uc5b4\ub5a4 \\(\\sim\\)\uac00 \uc874\uc7ac\ud55c\ub2e4&#8221; \ub610\ub294 &#8220;\\(\\cdots\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4&#8221;\ub97c \uc758\ubbf8\ud55c\ub2e4. \uc874\uc7ac\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud45c\ud604\ud55c \uba85\uc81c\ub97c <span class=\"defined\">\uc874\uc7ac\uba85\uc81c<\/span>(existential statement)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ub97c \ub4e4\uc5b4, &#8220;\uc81c\uacf1\ud558\uc5ec 2\uac00 \ub418\ub294 \uc2e4\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4&#8221;\ub77c\ub294 \ubb38\uc7a5\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[\\exists x \\in \\mathbb{R},\\,\\, x^2 = 2 \\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.10.<\/span><br \/>\n\ub2e4\uc74c \ud55c\uc815\uba85\uc81c\uac00 \ucc38\uc778\uc9c0 \uac70\uc9d3\uc778\uc9c0 \ud310\ub2e8\ud558\uace0 \uadf8 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\forall n\\in\\mathbb{Z},\\; n^2\\geq 0\\)<\/li>\n<li>\\(\\exists n\\in\\mathbb{Z},\\; n^2=2\\)<\/li>\n<li>\\(\\forall x\\in\\mathbb{R},\\; \\exists y\\in\\mathbb{R},\\; y>x\\)<\/li>\n<li>\\(\\exists y\\in\\mathbb{R},\\; \\forall x\\in\\mathbb{R},\\; y>x\\)<\/li>\n<\/ol>\n<\/div>\n<h4>\ud55c\uc815\uba85\uc81c\uc758 \ubd80\uc815<\/h4>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 2.2. (\ud55c\uc815\uba85\uc81c\uc758 \ubd80\uc815)<\/span><\/p>\n<p>\ud55c\uc815\uba85\uc81c\uc758 \ubd80\uc815\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\neg(\\forall x \\in A,\\, p(x)) &#038;\\,\\equiv\\, \\exists x \\in A,\\, \\neg p(x),\\\\[3pt]<br \/>\n\\neg(\\exists x \\in A,\\, q(x)) &#038;\\,\\equiv\\, \\forall x \\in A,\\, \\neg q(x).<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab \ubc88\uc9f8 \ub3d9\uce58\ub294 &#8220;\\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\(p(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4&#8221;\uac00 \uac70\uc9d3\uc774\ub77c\ub294 \uac83\uc774 \uace7 \\(p(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\ub294 \\(A\\)\uc758 \uc6d0\uc18c\uac00 \uc801\uc5b4\ub3c4 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4\ub294 \ub73b\uc774\ubbc0\ub85c \uc131\ub9bd\ud55c\ub2e4. \ub450 \ubc88\uc9f8 \ub3d9\uce58\ub294 &#8220;\\(q(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(A\\)\uc758 \uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4&#8221;\uac00 \uac70\uc9d3\uc774\ub77c\ub294 \uac83\uc774 \uace7 \\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\(q(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\ub294\ub2e4\ub294 \ub73b\uc774\ubbc0\ub85c \uc131\ub9bd\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.11.<\/span><br \/>\n\ub2e4\uc74c \uba85\uc81c\ub97c \ud55c\uc815\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b4\uace0, \uadf8 \ubd80\uc815\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ubaa8\ub4e0 \uc2e4\uc218\uc758 \uc81c\uacf1\uc740 \uc74c\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<li>\uc5b4\ub5a4 \uc790\uc5f0\uc218\ub294 \uc18c\uc218\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc591\uc758 \uc2e4\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec, \\(|x| < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(0\\)\uc774 \uc544\ub2cc \uc2e4\uc218 \\(x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h4>\uc9d1\ud569\uc758 \ud45c\ud604\uacfc \ud55c\uc815\uba85\uc81c<\/h4>\n<p>\uc870\uac74\uc81c\uc2dc\ubc95\uc73c\ub85c \uc9d1\ud569\uc744 \ub098\ud0c0\ub0bc \ub54c \ud55c\uc815\uba85\uc81c\uac00 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc740 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<ul>\n<li>\uc9dd\uc218\uc758 \uc9d1\ud569: \\(\\{n \\in \\mathbb{Z} \\mid \\exists k \\in \\mathbb{Z},\\, n = 2k\\}\\)<\/li>\n<li>\uc18c\uc218\uc758 \uc9d1\ud569: \\(\\{p \\in \\mathbb{N} \\mid p > 1 \\,\\text{ \uadf8\ub9ac\uace0 }\\, \\forall d \\in \\mathbb{N},\\, (d \\mid p \\to d = 1 \\,\\text{ \ub610\ub294 }\\, d = p)\\}\\)<\/li>\n<li>\uc720\ub9ac\uc218\uc758 \uc9d1\ud569: \\(\\{x \\in \\mathbb{R} \\mid \\exists p \\in \\mathbb{Z},\\, \\exists q \\in \\mathbb{Z} \\setminus \\{0\\},\\, x = \\frac{p}{q}\\}\\)<\/li>\n<\/ul>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-logic\/\">\uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n<p><span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/\">\uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch02-sets\">\uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\">\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\">\uad00\uacc4\uc640 \ud568\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">\uc790\uc5f0\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">\uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">\uc120\ud0dd \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch11-formal-logic\">\ud615\uc2dd\ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch13-soundness-completeness-proplogic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch18-peano-arithmetics\">\ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\">\ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uba85\uc81c\uc640 \ub17c\ub9ac \uc5f0\uc0b0, \uba85\uc81c \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774\uc81c \uc774\ub7ec\ud55c \ub17c\ub9ac\uc758 \uc5b8\uc5b4\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc218\ud559\uc758 \uae30\ubcf8 \ub300\uc0c1\uc778 \uc9d1\ud569\uc744 \ub2e4\ub8ec\ub2e4. 19\uc138\uae30 \ub9d0 \uce78\ud1a0\uc5b4(Georg Cantor)\uac00 \ubc1c\uc804\uc2dc\ud0a8 \uc9d1\ud569\ub860\uc740 \ud604\ub300 \uc218\ud559\uc758 \uac70\uc758 \ubaa8\ub4e0 \ubd84\uc57c\uc758 \uae30\ubc18\uc774 \ub418\uc5c8\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uc5f0\uc0b0\uc744 \ub2e4\ub8e8\uba70, \uba85\uc81c\uc640 \uc9d1\ud569 \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. 1. \uc9d1\ud569\uacfc \uc6d0\uc18c \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc9d1\ud569(set)\uc744 \uba85\ud655\ud558\uac8c \uad6c\ubcc4\ub418\ub294 \ub300\uc0c1\ub4e4\uc758 \ubaa8\uc784\uc73c\ub85c \uc9c1\uad00\uc801\uc73c\ub85c \uc774\ud574\ud55c\ub2e4. \uc9d1\ud569\uc744 \uc774\ub8e8\ub294 \uac01\uac01\uc758 \ub300\uc0c1\uc744 \uc6d0\uc18c(element)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9d1\ud569\uc740 \ubcf4\ud1b5 \ub300\ubb38\uc790&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":102,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9244","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9244","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=9244"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9244\/revisions"}],"predecessor-version":[{"id":10077,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9244\/revisions\/10077"}],"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=9244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}