{"id":2835,"date":"2019-03-05T23:40:14","date_gmt":"2019-03-05T14:40:14","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2835"},"modified":"2019-09-05T14:59:28","modified_gmt":"2019-09-05T05:59:28","slug":"calculus-sets-and-the-real-number-system","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-sets-and-the-real-number-system\/","title":{"rendered":"\uc9d1\ud569\uacfc \uc2e4\uc218"},"content":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubbf8\uc801\ubd84\ud559\uc744 \uacf5\ubd80\ud558\uae30 \uc704\ud574 \ud544\uc694\ud55c \uae30\ucd08 \uac1c\ub150\uc778 \uc218\ud559\uc758 \ub17c\ub9ac, \uc9d1\ud569\uc758 \uc131\uc9c8, \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8\uc744 \uac04\ub7b5\ud558\uac8c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc218\ud559\uc758 \ub17c\ub9ac<\/h3>\n<p>\uba3c\uc800 \ub17c\ub9ac\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub97c \uc0b4\ud3b4\ubcf4\uc790.<!-- \/p --><\/p>\n<p><!-- div class=\"definition\" --><br \/>\n<!-- p><span class=\"definition\">\uc815\uc758 1. (\ub17c\ub9ac \uae30\ud638)<\/span><\/p -->\n<!-- p -->\\(p\\)\uc640 \\(q\\)\uac00 \uc218\ud559\uc801 \ubb38\uc7a5\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\ub17c\ub9ac\uacf1<\/span> :&nbsp; \u2018\\(p\\) \uadf8\ub9ac\uace0 \\(q\\)\u2019\ub97c \\((p \\,\\wedge\\, q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((p \\,\\wedge\\, q)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \\(p\\)\uc640 \\(q\\)\uac00 \ubaa8\ub450 \ucc38\uc774\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\ud569<\/span>:&nbsp; \u2018\\(p\\) \ub610\ub294 \\(q\\)\u2019\ub97c \\((p\\,\\vee\\,q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((p\\,\\vee\\,q)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \\(p\\)\uc640 \\(q\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \ucc38\uc774\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\uc870\uac74\ubb38<\/span>:&nbsp; \u2018\\(p\\)\uc774\uba74 \\(q\\)\uc774\ub2e4\u2019\ub97c \\((p\\,\\rightarrow \\,q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((p\\,\\rightarrow \\,q)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \u2018\\(p\\)\uac00 \ucc38\uc77c \ub54c\ub9c8\ub2e4 \\(q\\)\uac00 \ucc38\uc774\ub2e4\u2019\ub97c \uc758\ubbf8\ud55c\ub2e4. \ud2b9\ud788 \\(p\\)\uac00 \ud56d\uc0c1 \uac70\uc9d3\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(q\\)\uc758 \uc9c4\ub9ac \uc5ec\ubd80\uc5d0 \uc0c1\uad00 \uc5c6\uc774 \\((p\\,\\rightarrow \\,q)\\)\ub294 \ucc38\uc774\ub2e4.<\/li>\n<li><span class=\"defined\">\ubd80\uc815<\/span>:&nbsp; \u2018\\(p\\)\uac00 \uc544\ub2c8\ub2e4\u2019\ub97c \\((\\neg p)\\) \ub610\ub294 \\((\\sim p)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((\\neg p)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \\(p\\)\uac00 \uac70\uc9d3\uc774\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\uc804\uce6d\uae30\ud638<\/span>:&nbsp; \u2018\\(A\\)\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \\(x\\)\uac00 \\(p\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\u2019\ub97c \\((\\forall x \\in A) p\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(p\\)\uac00 \\(x\\)\uc758 \ud568\uc218\ub77c\ub294 \uac83\uc744 \uac15\uc870\ud558\uc5ec \\(p(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4\uba74, \\((\\forall x \\in A) p\\)\ub97c \\((\\forall x \\in A) p(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li><span class=\"defined\">\uc874\uc7ac\uae30\ud638<\/span>:&nbsp; \u2018\\(A\\)\uc5d0 \uc18d\ud558\ub294 \\(x\\) \uc911\uc5d0\uc11c \\(p\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4\u2019\ub97c \\((\\exists x \\in A) p\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(p\\)\uac00 \\(x\\)\uc758 \ud568\uc218\ub77c\ub294 \uac83\uc744 \uac15\uc870\ud558\uc5ec \\(p(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4\uba74, \\((\\exists x \\in A) p\\)\ub97c \\((\\exists x \\in A) p(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<p><!-- \/div --><\/p>\n<p>\uc804\uce6d\uae30\ud638\uc640 \uc874\uc7ac\uae30\ud638\ub97c \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\ud55c\uc815\uae30\ud638<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1. (\ub17c\ub9ac\uc2dd\uc758 \ud45c\ud604)<\/span><\/p>\n<ul>\n<li>\u2018\\(x\\)\ub294 \uc0bc\uac01\ud615\uc774\ub2e4\u2019\ub97c \\(p\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \u2018\\(x\\)\ub294 \ub450 \ubcc0\uc758 \uae38\uc774\uac00 \uac19\uc740 \ub2e4\uac01\ud615\uc774\ub2e4\u2019\ub97c \\(q\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \u2018\\(x\\)\ub294 \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615\uc774\ub2e4\u2019\ub294 \\((p\\,\\wedge\\,q)\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/li>\n<li>\u2018\\(x\\)\ub294 \uc18c\uc218\uc774\ub2e4\u2019\ub97c \\(p\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \u2018\\(x\\)\ub294 \ud569\uc131\uc218\uc774\ub2e4\u2019\ub97c \\(q\\)\ub85c \ub098\ud0c0\ub0b4\uba70 \u2018\\(x\\)\ub294 \\(1\\)\uc774\ub2e4\u2019\ub97c \\(r\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \\(\\neg p\\)\uc640 \\((q \\vee r)\\)\ub294 \uac19\uc740 \uc758\ubbf8\ub97c \uac16\ub294 \ub17c\ub9ac\uc2dd\uc774\ub2e4. (\uac19\uc740 \uc758\ubbf8\ub97c \uac16\ub294\ub2e4\ub294 \uac83\uc740 \ubcc0\uc218 \\(x\\)\uc5d0 \uc5b4\ub290 \uc0bc\uac01\ud615\uc744 \ub300\uc785\ud558\ub4e0 \\((\\neg p)\\)\uc640 \\((q\\,\\vee\\,r)\\)\uac00 \ub3d9\uc77c\ud55c \uc9c4\ub9bf\uac12\uc744 \uac16\ub294\ub2e4\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.)<\/li>\n<\/ul>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2. (\ud55c\uc815\uae30\ud638)<\/span><\/p>\n<p>\ub9cc\uc57d \u2018\\(x\\)\ub294 \uc815\uc0bc\uac01\ud615\uc774\ub2e4\u2019\ub97c \\(p(x)\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \u2018\\(x\\)\ub294 \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615\uc774\ub2e4\u2019\ub97c \\(q(x)\\)\ub85c \ub098\ud0c0\ub0b4\uba70 \ubaa8\ub4e0 \uc0bc\uac01\ud615\uc774 \uc9d1\ud569\uc744 \\(T\\)\ub77c\uace0 \ud55c\ub2e4\uba74 \u201c\uc815\uc0bc\uac01\ud615\uc740 \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615\uc774\ub2e4\u201d\ub294<br \/>\n\\[(\\forall x\\in T)(p(x) \\,\\rightarrow\\, q(x))\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub610 \u201c\uc774\ub4f1\ubcc0\uc0bc\uac01\ud615 \uc911\uc5d0\uc11c\ub294 \uc815\uc0bc\uac01\ud615\uc774 \uc544\ub2cc \uac83\uc774 \uc788\ub2e4\u201d\ub294<br \/>\n\\[(\\exists x \\in T)( ( q(x)\\,\\wedge\\, \\neg p(x)) )\\tag{1}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ud55c\ud3b8 (1)\uc740 \u201c\ubaa8\ub4e0 \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615\uc740 \uc815\uc0bc\uac01\ud615\uc774\ub2e4\u201d\uc758 \ubd80\uc815\uacfc \uac19\uc73c\ubbc0\ub85c<br \/>\n\\[(\\neg (\\forall x\\in T) (q(x)\\,\\rightarrow\\,p(x)))\\tag{2}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4. \uc989 (1)\uacfc (2)\ub294 \uac19\uc740 \ub73b\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\\(p\\)\uc640 \\(q\\)\uac00 \uc218\ud559\uc801 \ubb38\uc7a5\uc77c \ub54c \\((p\\,\\rightarrow\\,q)\\)\uc640 \\((q\\,\\rightarrow\\,p)\\)\uac00 \ubaa8\ub450 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \\((p\\,\\leftrightarrow\\,q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud55c\ud3b8 \\((p\\,\\rightarrow\\,q)\\)\uac00 \ucc38\uc778 \uac83\uc744 \\((p\\,\\Rightarrow\\,q)\\)\ub85c \ub098\ud0c0\ub0b4\uace0, \\((p\\,\\leftrightarrow\\,q)\\)\uac00 \ucc38\uc778 \uac83\uc744 \\((p\\,\\Leftrightarrow\\,q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ud568\uc758 \\((p\\,\\rightarrow\\,q)\\)\uc640 \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\((p\\,\\rightarrow\\,q)\\)\uc758 <span class=\"defined\">\uc5ed<\/span>:&nbsp; \\((q\\,\\rightarrow\\,p)\\)<\/li>\n<li>\\((p\\,\\rightarrow\\,q)\\)\uc758 <span class=\"defined\">\uc774<\/span>:&nbsp; \\(((\\neg p)\\,\\rightarrow\\,(\\neg q))\\)<\/li>\n<li>\\((p\\,\\rightarrow\\,q)\\)\uc758 <span class=\"defined\">\ub300\uc6b0<\/span>:&nbsp; \\(((\\neg q)\\,\\rightarrow\\,(\\neg p))\\)<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3. (\uc870\uac74\ubb38\uc758 \uc5ed, \uc774, \ub300\uc6b0)<\/span><\/p>\n<p>\ub300\uc0c1 \uc601\uc5ed\uc774 \uc2e4\uc218 \uc804\uccb4 \uc9d1\ud569\uc77c \ub54c, \u2018\\(x > 3\\)\uc774\uba74 \\(x > 2\\)\uc774\ub2e4\u2019\ub77c\ub294 \ubb38\uc7a5\uc758 \uc5ed, \uc774 \ub300\uc6b0\ub97c \uad6c\ud574\ubcf4\uc790.<\/p>\n<ul>\n<li>\uc5ed&nbsp;:&nbsp; \u2018\\(x > 2\\)\uc774\uba74 \\(x > 3\\)\uc774\ub2e4.\u2019<\/li>\n<li>\uc774&nbsp;:&nbsp; \u2018\\(x \\le 3\\)\uc774\uba74 \\(x \\le 2\\)\uc774\ub2e4.\u2019<\/li>\n<li>\ub300\uc6b0&nbsp;:&nbsp; \u2018\\(x \\le 2\\)\uc774\uba74 \\(x \\le 3\\)\uc774\ub2e4.\u2019<\/li>\n<\/ul>\n<\/div>\n<p>\\(x\\)\ub97c \uacf5\ud1b5 \ubcc0\uc218\ub85c \uac16\ub294 \ub450 \ubb38\uc7a5 \\(p(x)\\)\uc640 \\(q(x)\\)\uac00 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc11c\ub85c \uac19\uc740 \uc9c4\ub9bf\uac12(\ucc38, \uac70\uc9d3)\uc744 \uac00\uc9c8 \ub54c \u2018\\(p\\)\uc640 \\(q\\)\ub294 \uc11c\ub85c <span class=\"defined\">\ub3d9\ub4f1\ud558\ub2e4<\/span>\u2019 \ub610\ub294 \u2018\\(p\\)\uc640 \\(q\\)\ub294 \uc11c\ub85c <span class=\"defined\">\ub3d9\uce58\uc774\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294 \u2018\\( p \\,\\equiv\\, q \\)\u2019\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uadf8\ub7f0\ub370 \\((p\\,\\Leftrightarrow\\,q)\\)\uc77c \ub54c\uc5d0\ub3c4 \\(p(x)\\)\uc640 \\(q(x)\\)\ub294 \uc11c\ub85c \uac19\uc740 \uc9c4\ub9bf\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c \u2018\\( p \\,\\equiv\\, q \\)\u2019\ub97c \u2018\\( p \\,\\Leftrightarrow\\, q \\)\u2019\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \ub17c\ub9ac\uc2dd\uc5d0\uc11c \\(\\equiv\\) \uae30\ud638\uc640 \\(\\Leftrightarrow\\)\ub294 \ub4f1\uc2dd\uc758 \ub4f1\ud638\uc640 \uac19\uc740 \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n<p>\uc608\ucee8\ub300 \\(p\\)\uc640 \\(q\\) \uac01\uac01\uc774 \uc5b4\ub5a0\ud55c \uc9c4\ub9bf\uac12\uc744 \uac16\ub4e0 \\((p \\,\\rightarrow\\,q)\\)\uc640 \\(((\\neg p)\\,\\vee\\,q)\\)\ub294 \uac19\uc740 \uc9c4\ub9bf\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c<br \/>\n\\[(p\\,\\rightarrow\\,q) \\,\\,\\Longleftrightarrow \\,\\, ((\\neg p)\\,\\vee\\,q)\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\uc774 \ubc16\uc5d0 \uc218\ud559\uc758 \ub17c\ub9ac\uc5d0\uc11c\ub294 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \ubc95\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc774\uc911\ubd80\uc815 \ubc95\uce59<\/span> :&nbsp; \\((\\neg (\\neg p)) \\,\\, \\Longleftrightarrow \\,\\, p \\)<\/li>\n<li><span class=\"defined\">\ud569\uc758 \ubc95\uce59<\/span> :&nbsp; \\( p \\,\\, \\Longrightarrow \\,\\, (p \\vee q) \\)<\/li>\n<li><span class=\"defined\">\ub2e8\uc21c\ud654 \ubc95\uce59<\/span> :&nbsp; \\((p \\wedge q) \\,\\, \\Longrightarrow \\,\\, p \\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\ud569\uc758 \uba71\ub4f1 \ubc95\uce59<\/span> :&nbsp; \\((p \\vee p) \\,\\,\\Longleftrightarrow \\,\\, p\\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\uacf1\uc758 \uba71\ub4f1 \ubc95\uce59<\/span> :&nbsp; \\((p \\wedge p) \\,\\,\\Longleftrightarrow \\,\\, p\\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\ud569\uc758 \uad50\ud658 \ubc95\uce59<\/span> :&nbsp; \\((p \\vee q) \\,\\, \\Longleftrightarrow \\,\\, (q \\vee p) \\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\uacf1\uc758 \uad50\ud658 \ubc95\uce59<\/span> :&nbsp; \\((p \\wedge q) \\,\\, \\Longleftrightarrow \\,\\, (q \\wedge p) \\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\ud569\uc758 \uacb0\ud569 \ubc95\uce59<\/span> :&nbsp; \\((p \\vee (q \\vee r )) \\,\\, \\Longleftrightarrow \\,\\, ((p \\vee q) \\vee r) \\)<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\uacf1\uc758 \uacb0\ud569 \ubc95\uce59<\/span> :&nbsp; \\((p \\wedge (q \\wedge r )) \\,\\, \\Longleftrightarrow \\,\\, ((p \\wedge q) \\wedge r) \\)<\/li>\n<li><span class=\"defined\">\ubd84\ubc30 \ubc95\uce59<\/span> :&nbsp; \\((p \\vee (q \\wedge r)) \\,\\, \\Longleftrightarrow \\,\\, ((p \\vee q) \\wedge (p \\vee r)) \\)<\/li>\n<li><span class=\"defined\">\ubd84\ubc30 \ubc95\uce59<\/span> :&nbsp; \\((p \\wedge (q \\vee r)) \\,\\, \\Longleftrightarrow \\,\\, ((p \\wedge q) \\vee (p \\wedge r)) \\)<\/li>\n<li><span class=\"defined\">\ucd94\uc774\ubc95\uce59<\/span> :&nbsp; \\(((p \\,\\rightarrow\\,q) \\,\\wedge\\, (q\\,\\rightarrow\\,r)) \\,\\, \\Longrightarrow \\,\\, (p\\,\\rightarrow\\,q) \\)<\/li>\n<li><span class=\"defined\">\ub300\uc6b0\ubc95\uce59<\/span> :&nbsp; \\((p\\,\\rightarrow\\,q) \\,\\,\\Longleftrightarrow \\,\\, ((\\neg q)\\,\\rightarrow\\,(\\neg p))\\)<\/li>\n<li><span class=\"defined\">\ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span> :&nbsp; \\((\\neg(p\\,\\wedge\\,q)) \\,\\,\\Longleftrightarrow\\,\\,((\\neg p) \\,\\vee\\, (\\neg q))\\)<\/li>\n<li><span class=\"defined\">\ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span> :&nbsp; \\((\\neg(p\\,\\vee\\,q)) \\,\\,\\Longleftrightarrow\\,\\,((\\neg p) \\,\\wedge\\, (\\neg q))\\)<\/li>\n<li><span class=\"defined\">\uc804\uce6d\uba85\uc81c\uc758 \ubd80\uc815<\/span> :&nbsp; \\((\\neg (\\forall x \\in A) p(x)) \\,\\, \\Longleftrightarrow \\,\\, ((\\exists x\\in A)(\\neg p(x)))\\)<\/li>\n<li><span class=\"defined\">\uc874\uc7ac\uba85\uc81c\uc758 \ubd80\uc815<\/span> :&nbsp; \\((\\neg (\\exists x \\in A) p(x)) \\,\\, \\Longleftrightarrow \\,\\, ((\\forall x\\in A)(\\neg p(x)))\\)<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4. (\uc870\uac74\ubb38\uc758 \uc815\uc758\uc640 \uc131\uc9c8\uc744 \uc774\uc6a9\ud55c \uc99d\uba85)<\/span><\/p>\n<p>\uacf5\uc9d1\ud569\uc774 \uc784\uc758\uc758 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\ud798\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\\(\\varnothing\\)\uc774 \uacf5\uc9d1\ud569\uc774\uace0 \\(A\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(\\varnothing \\subseteq A\\)\ub97c \uc99d\uba85\ud558\ub824\uba74<br \/>\n\\[((x \\in \\varnothing) \\,\\, \\rightarrow \\,\\, (x\\in A))\\]<br \/>\n\uac00 \ucc38\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \uc704 \ubb38\uc7a5\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[((\\neg(x \\in \\varnothing)) \\, \\vee \\, (x\\in A))\\]<br \/>\n\uc989<br \/>\n\\[((x \\notin \\varnothing ) \\, \\vee \\, (x\\in A)).\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\varnothing\\)\ub294 \uc5b4\ub5a0\ud55c \uc6d0\uc18c\ub3c4 \uac00\uc9c0\uace0 \uc788\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\((x \\notin \\varnothing )\\)\uc740 \ucc38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\in A\\)\uc758 \uc9c4\ub9bf\uac12\uc774 \ubb34\uc5c7\uc774\ub4e0 \uc0c1\uad00 \uc5c6\uc774 \uc704 \ubb38\uc7a5 \uc804\uccb4\ub294 \ucc38\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\ud55c\uc815\uae30\ud638\uac00 \ub450 \uac1c \uc774\uc0c1 \uc5f0\ub2ec\uc544 \uc0ac\uc6a9\ub418\ub294 \uacbd\uc6b0\ub3c4 \uc788\ub2e4. \uc608\ucee8\ub300 \uc790\uc5f0\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744 \\(\\mathbb{N}\\)\uc774\ub77c\uace0 \ud558\uba74 \u2018\uc784\uc758\uc758 \uc790\uc5f0\uc218\uc5d0 \ub300\ud558\uc5ec \uadf8\ubcf4\ub2e4 \ub354 \ud070 \uc790\uc5f0\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4\u2019\ub77c\ub294 \ubb38\uc7a5\uc740<br \/>\n\\[(\\forall n \\in \\mathbb{N})(\\exists m \\in \\mathbb{N}) (n < m)\\tag{3}\\]\n\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \ub450 \ud55c\uc815\uae30\ud638\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4 \uc4f0\uba74 \uc758\ubbf8\uac00 \ub2ec\ub77c\uc9c4\ub2e4. \uc989\n\\[(\\exists m \\in \\mathbb{N})(\\forall n \\in \\mathbb{N}) (n < m)\\tag{4}\\]\n\uc740 (3)\uacfc\ub294 \ub2e4\ub978 \uc758\ubbf8\uc774\ub2e4. \uc989 (4)\ub294 \u2018\uc5b4\ub5a0\ud55c \uc790\uc5f0\uc218\uac00 \uc874\uc7ac\ud558\uc5ec \uadf8 \uc790\uc5f0\uc218\uac00 \ubaa8\ub4e0 \uc790\uc5f0\uc218\ubcf4\ub2e4 \ub354 \ud06c\ub2e4\u2019\ub77c\ub294 \ub73b\uc774\ub2e4. (3)\uc740 \ucc38\uc774\uc9c0\ub9cc (4)\ub294 \uac70\uc9d3\uc774\ub2e4.<\/p>\n<p>\ud55c\uc815\uae30\ud638\uac00 \ub450 \uac1c \uc774\uc0c1 \uc0ac\uc6a9\ub41c \uc2dd\uc744 \uc790\uc5f0\uc5b8\uc5b4\ub85c \ud45c\ud604\ud558\uba74 \uc758\ubbf8\uac00 \ubaa8\ud638\ud574\uc9c0\ub294 \uacbd\uc6b0\uac00 \uc788\ub2e4. \uc608\ucee8\ub300 \u201c\uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(n < m\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\u201d\ub77c\ub294 \ubb38\uc7a5\uc740 \uad00\uc810\uc5d0 \ub530\ub77c\uc11c (3)\uc744 \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\uace0 (4)\ub97c \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ubaa8\ud638\ud568\uc744 \ud574\uacb0\ud558\ub294 \ubc29\ubc95\uc740 \uba87 \uac00\uc9c0\uac00 \uc788\ub294\ub370, \uadf8 \uc911 \ud558\ub098\ub294 \uad04\ud638\ub97c \uc774\uc6a9\ud558\ub294 \uac83\uc774\ub2e4. \uc989 (3)\uc744 \ub098\ud0c0\ub0b4\uae30 \uc704\ud574\uc11c\ub294 \u201c\uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec [\\(n < m\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac]\ud55c\ub2e4\u201d\ub77c\uace0 \uc4f0\uba74 \ub418\uace0, (4)\ub97c \ub098\ud0c0\ub0b4\uae30 \uc704\ud574\uc11c\ub294 \u201c[\uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(n < m\\)]\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\u201d\ub77c\uace0 \uc4f0\uba74 \ub41c\ub2e4.<\/p>\n<p>\uad04\ud638\ub97c \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc740 \uad6c\uc5b4\uc5d0\uc11c\ub294 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \uc774\ub54c\uc5d0\ub294 \ud55c\uc815\uae30\ud638\ub97c \uc4f0\ub294 \uc21c\uc11c\ub300\ub85c \ub9d0\ub85c \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4. \uc989 (3)\uc744 \ub098\ud0c0\ub0b4\uae30 \uc704\ud574\uc11c\ub294 \u201c\uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(n\\)\uc774 \\(m\\)\ubcf4\ub2e4 \uc791\ub2e4\u201d\ub77c\uace0 \ud558\uba74 \ub418\uace0, (4)\ub97c \ub098\ud0c0\ub0b4\uae30 \uc704\ud574\uc11c\ub294 \u201c\uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \\(n\\)\uc774 \\(m\\)\ubcf4\ub2e4 \uc791\ub2e4\u201d\ub77c\uace0 \ud558\uba74 \ub41c\ub2e4. \uad6c\uc5b4\uac00 \uc624\ud788\ub824 \ub354 \uad6c\uc5b4 \uac19\uc9c0 \uc54a\uac8c \ub290\uaef4\uc9c0\ub294 \uc544\uc774\ub7ec\ub2c8\ub294 \ub364\uc774\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc9d1\ud569\uc758 \uc5f0\uc0b0<\/h3>\n<p>\uba85\ud655\ud55c \uae30\uc900\uc5d0 \ub530\ub77c \uc218\ud559\uc801 \ub300\uc0c1\uc744 \ubaa8\uc544\ub193\uc740 \uac83\uc744 <span class=\"defined\">\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ucee8\ub300 \u2018\uc9dd\uc218\uc778 \ubaa8\ub4e0 \uc790\uc5f0\uc218\uc758 \ubaa8\uc784\u2019, \u2018\uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \uc2e4\uc218 \uc804\uccb4 \uc9d1\ud569\uc774\uace0 \uc720\uacc4\uc778 \ud568\uc218\uc758 \ubaa8\uc784\u2019, \u2018\\(\\pi\\)\uc758 \uc2ed\uc9c4 \uc804\uac1c\uc5d0\uc11c \uc18c\uc218\uc810 \uc544\ub798 \uc22b\uc790 \ubc30\uc5f4\ub85c \ub098\ud0c0\ub098\ub294 \uc790\uc5f0\uc218\uc758 \ubaa8\uc784\u2019\uc740 \ubaa8\ub450 \uc9d1\ud569\uc774\ub2e4. (\ubb3c\ub860 \uc5b4\ub5a0\ud55c \uc790\uc5f0\uc218\uac00 \\(\\pi\\)\uc758 \uc2ed\uc9c4 \uc804\uac1c\uc5d0\uc11c \uc18c\uc218\uc810 \uc544\ub798 \uc22b\uc790 \ubc30\uc5f4\ub85c \ub098\ud0c0\ub098\ub294\uc9c0 \uc54c\uc544\ub0b4\ub294 \uacf5\uc2dd\uc740 \uc54c\ub824\uc838 \uc788\uc9c0 \uc54a\uc9c0\ub9cc.) \ubc18\uba74 \u2018\ub9e4\uc6b0 \ud070 \uc218\uc758 \ubaa8\uc784\u2019(\ud070 \uc218\uc758 \uae30\uc900\uc774 \uc5c6\uc74c), \u2018\uc5f0\uc18d\ud568\uc218\uc758 \ubaa8\uc784\u2019(\uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \uba85\ud655\ud558\uac8c \uc815\ud574\uc9c0\uc9c0 \uc54a\uc74c), \u2018\ub0b4\uac00 \uc88b\uc544\ud558\ub294 \uc0ac\ub78c\ub4e4\uc758 \ubaa8\uc784\u2019(\ub54c\uc5d0 \ub530\ub77c \ubc14\ub014 \uc218 \uc788\uc74c)\uc740 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<p>\uc9d1\ud569\uc744 \uad6c\uc131\ud558\uace0 \uc788\ub294 \ub300\uc0c1 \uac01\uac01\uc744 \uadf8 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub77c\uace0 \ubd80\ub978\ub2e4. \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc758 \uc6d0\uc18c\uc778 \uac83\uc744 \\(a\\in A\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \u2018\\(a\\)\uac00 \\(A\\)\uc5d0 \uc18d\ud55c\ub2e4\u2019\ub77c\uace0 \uc77d\ub294\ub2e4. \uc608\ucee8\ub300 \uc9dd\uc218\uc778 \ubaa8\ub4e0 \uc790\uc5f0\uc218\uc758 \ubaa8\uc784\uc744 \\(E\\)\ub77c\uace0 \ud558\uba74 \\(8\\in E\\)\uc774\uc9c0\ub9cc \\(5 \\notin E\\)\uc774\ub2e4. \ubaa8\uc21c\uc744 \ud53c\ud558\uae30 \uc704\ud558\uc5ec \uc9d1\ud569\uc740 \uc790\uae30 \uc790\uc2e0\uc758 \uc6d0\uc18c\uac00 \uc544\ub2cc \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \u2018\ubaa8\ub4e0 \uc9d1\ud569\uc758 \ubaa8\uc784\u2019\uc740 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \ubaa8\ub4e0 \uc9d1\ud569\uc758 \ubaa8\uc784 \\(S\\)\uac00 \uc9d1\ud569\uc774\ub77c\uba74 \\(S \\in S\\)\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\uc6d0\uc18c\uac00 \ud558\ub098\ub3c4 \uc5c6\ub294 \ubaa8\uc784\ub3c4 \uc9d1\ud569\uc73c\ub85c \uac04\uc8fc\ud558\uc5ec <span class=\"defined\">\uacf5\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uacf5\uc9d1\ud569\uc740 \uae30\ud638\ub85c \\(\\varnothing\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc9d1\ud569\uc744 \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\ub2e4. \uba3c\uc800 \u2018\u2026\uc778 \u2026\uc758 \ubaa8\uc784\uc744 \\(E\\)\ub77c\uace0 \ud558\uc790\u2019\uc640 \uac19\uc740 \uaf34\ub85c \uc9d1\ud569\uc744 \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4. \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \uc801\uc5b4\uc11c \ub2e4 \ub098\uc5f4\ud560 \uc218 \uc788\uac70\ub098, \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \ub9ce\uc544\ub3c4 \ubb38\ub9e5\uc0c1 \uba85\ud655\ud558\uac8c \uc778\uc9c0\ud560 \uc218 \uc788\ub294 \uaddc\uce59\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4\uba74<br \/>\n\\[A = \\left\\{ 2,\\,4,\\,6,\\,\\cdots,\\,20 \\right \\}\\]<br \/>\n\ub610\ub294<br \/>\n\\[B = \\left\\{ 3,\\,6,\\,9,\\,12,\\,15,\\, \\cdots \\right\\}\\]<br \/>\n\uc640 \uac19\uc774 \uc6d0\uc18c\ub97c \ub098\uc5f4\ud558\uc5ec \uc9d1\ud569\uc744 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ubc29\ubc95\uc744 <span class=\"defined\">\uc6d0\uc18c\ub098\uc5f4\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud55c\ud3b8 \uc704 \ub450 \uc9d1\ud569\uc740<br \/>\n\\[\\begin{align}<br \/>\nA &#038;= \\left\\{ x \\in \\mathbb{N} \\,\\vert\\, x \\text{ is an even number less than or equal to 20} \\right\\}, \\\\[8pt]<br \/>\nB &#038;= \\left\\{ x \\in \\mathbb{N} \\,\\vert\\, x \\text{ is divisible by 3} \\right\\}<br \/>\n\\end{align}\\]<br \/>\n\uacfc \uac19\uc774<br \/>\n\\[E = \\left\\{ x \\in U \\,\\vert\\, p(x) \\right\\}\\]<br \/>\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294\ub370, \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc744 <span class=\"defined\">\uc870\uac74\uc81c\uc2dc\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(A\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uc9d1\ud569 \\(B\\)\uc5d0 \uc18d\ud560 \ub54c \u2018\\(A\\)\ub294 \\(B\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc9d1\ud569<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \\(A \\subseteq B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(A\\subseteq B\\)\uc774\uba74\uc11c \ub3d9\uc2dc\uc5d0 \\(B\\subseteq A\\)\uc77c \ub54c, \uc989 \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\)\ub97c \uad6c\uc131\ud558\ub294 \uc6d0\uc18c\uac00 \uc11c\ub85c \ub3d9\uc77c\ud560 \ub54c, \u2018\\(A\\)\uc640 \\(B\\)\ub294 <span class=\"defined\">\uc11c\ub85c \uac19\ub2e4<\/span>\u2019\ub77c\uace0 \ud558\uace0 \\(A=B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(A \\subseteq B\\)\uc774\uc9c0\ub9cc \\(A \\ne B\\)\uc77c \ub54c \u2018\\(A\\)\ub294 \\(B\\)\uc758 <span class=\"defined\">\uc9c4\ubd80\ubd84\uc9d1\ud569<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \\(A\\subsetneq B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \ubaa8\ub450\uc5d0 \uc18d\ud55c \uc6d0\uc18c\ub4e4\uc758 \ubaa8\uc784\uc744 \\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\uad50\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A \\cap B \\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub450 \uc9d1\ud569 \\(A\\)\uc640 \\(B\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc5d0 \uc18d\ud55c \uc6d0\uc18c\ub4e4\uc758 \ubaa8\uc784\uc744 \\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\ud569\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A\\cup B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0\ub9cc \uc9d1\ud569 \\(B\\)\uc5d0\ub294 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub4e4\uc758 \ubaa8\uc784\uc744 \\(A\\)\ub85c\ubd80\ud130 \\(B\\)\uc758 <span class=\"defined\">\ucc28\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A\\setminus B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989<br \/>\n\\[\\begin{align}<br \/>\nA\\cap B &#038;= \\left\\{ x \\,\\vert\\, x \\in A \\,\\,\\wedge\\,\\, x\\in B \\right \\}, \\\\[8pt]<br \/>\nA\\cup B &#038;= \\left\\{ x \\,\\vert\\, x \\in A \\,\\,\\vee\\,\\, x\\in B \\right \\}, \\\\[8pt]<br \/>\nA \\setminus B &#038;= \\left\\{ x \\,\\vert\\, x \\in A \\,\\,\\wedge\\,\\, x\\notin B \\right \\}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(\\mathcal{U}\\)\ub97c \uae30\uc900\uc774 \ub418\ub294 \uc9d1\ud569\uc73c\ub85c \ub450\uc5c8\uc744 \ub54c, \\(\\mathcal{U}\\)\uc758 \uc6d0\uc18c \uc911 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uac83\ub4e4\uc758 \ubaa8\uc784\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc5ec\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A^C\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989<br \/>\n\\[A^C = \\left\\{ x\\in \\mathcal{U} \\,\\vert\\, x \\notin A \\right\\}\\]<br \/>\n\uc774\ub2e4. \uc774\ub54c \\(\\mathcal{U}\\)\ub97c <span class=\"defined\">\uc804\uccb4\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uc9d1\ud569\uc740 \uc804\uccb4\uc9d1\ud569\uc774 \uc815\ud574\uc838 \uc788\uc744 \ub54c\uc5d0\ub9cc \uc758\ubbf8\ub97c \uac00\uc9c4\ub2e4.<\/p>\n<p>\uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uc9d1\ud569\uc758 \uc5f0\uc0b0 \ubc95\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc774\uc911\ubd80\uc815 \ubc95\uce59<\/span> :&nbsp; \\( \\left( A ^C \\right) ^C = A \\)<\/li>\n<li><span class=\"defined\">\ud569\uc758 \ubc95\uce59<\/span> :&nbsp; \\(A \\subseteq A \\cup B \\)<\/li>\n<li><span class=\"defined\">\ub2e8\uc21c\ud654 \ubc95\uce59<\/span> :&nbsp; \\(A \\cap B \\subseteq A\\)<\/li>\n<li><span class=\"defined\">\ud569\uc9d1\ud569\uc758 \uba71\ub4f1 \ubc95\uce59<\/span> :&nbsp; \\(A \\cup A = A\\)<\/li>\n<li><span class=\"defined\">\uad50\uc9d1\ud569\uc758 \uba71\ub4f1 \ubc95\uce59<\/span> :&nbsp; \\(A \\cap A = A\\)<\/li>\n<li><span class=\"defined\">\ud569\uc9d1\ud569\uc758 \uad50\ud658 \ubc95\uce59<\/span> :&nbsp; \\(A \\cup B = B \\cup A\\)<\/li>\n<li><span class=\"defined\">\uad50\uc9d1\ud569\uc758 \uad50\ud658 \ubc95\uce59<\/span> :&nbsp; \\(A \\cap B = B \\cap A\\)<\/li>\n<li><span class=\"defined\">\ud569\uc9d1\ud569\uc758 \uacb0\ud569 \ubc95\uce59<\/span> :&nbsp; \\(A \\cup (B \\cup C ) = (A \\cup B ) \\cup C \\)<\/li>\n<li><span class=\"defined\">\uad50\uc9d1\ud569\uc758 \uacb0\ud569 \ubc95\uce59<\/span> :&nbsp; \\(A \\cap (B \\cap C ) = (A \\cap B ) \\cap C \\)<\/li>\n<li><span class=\"defined\">\ubd84\ubc30 \ubc95\uce59<\/span> :&nbsp; \\(A \\cup (B \\cap C ) = (A \\cup B) \\cap (A \\cup C) \\)<\/li>\n<li><span class=\"defined\">\ubd84\ubc30 \ubc95\uce59<\/span> :&nbsp; \\(A \\cap (B \\cup C ) = (A \\cap B) \\cup (A \\cap C) \\)<\/li>\n<li><span class=\"defined\">\ucd94\uc774 \ubc95\uce59<\/span> :&nbsp; \\( (A \\subseteq B \\,\\, \\wedge \\,\\, B \\subseteq C) \\,\\, \\Longrightarrow \\,\\, A \\subseteq C \\)<\/li>\n<li><span class=\"defined\">\ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span> :&nbsp; \\((A\\cap B)^C = A^C \\cup B^C\\)<\/li>\n<li><span class=\"defined\">\ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59<\/span> :&nbsp; \\((A\\cup B)^C = A^C \\cap B^C\\)<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5. (\uc9d1\ud569\uc758 \ud3ec\ud568\uad00\uacc4 \uc99d\uba85)<\/span><\/p>\n<p>\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc77c \ub54c \\(A\\cap B \\subseteq A\\cup B\\)\uc784\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc774 \ud3ec\ud568\uad00\uacc4\ub97c \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc9c4\uc220\uc774 \ucc38\uc784\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4.<br \/>\n\\[ (x\\in (A\\cap B)) \\,\\rightarrow\\, (x\\in (A\\cup B))\\]<br \/>\n\uc774\uac83\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uacfc\uc815\uc744 \ud1b5\ud574 \ubcf4\uc77c \uc218 \uc788\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n(x\\in (A\\cap B))<br \/>\n&#038; \\,\\Rightarrow\\, ((x\\in A) \\,\\wedge\\, (x\\in B)) \\\\[8pt]<br \/>\n&#038; \\,\\Rightarrow\\, (x\\in A) \\\\[8pt]<br \/>\n&#038; \\,\\Rightarrow\\, ((x\\in A) \\,\\vee\\, (x\\in B)) \\\\[8pt]<br \/>\n&#038; \\,\\Rightarrow\\, (x\\in (A\\cup B)).<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(A\\cap B\\)\ub294 \\(A\\cup B\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uad50\uc9d1\ud569\uacfc \ud569\uc9d1\ud569\uc740 \uc5ec\ub7ec \uac1c\uc758 \uc9d1\ud569\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc815\uc758\ub41c\ub2e4. \uc989 \\(A_1 ,\\) \\(A_2 ,\\) \\(\\cdots ,\\) \\(A_n\\)\uc774 \uc9d1\ud569\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\n\\bigcap_{i=1}^{n} A_i &#038;= A_1 \\cap A_2 \\cap \\cdots \\cap A_n , \\\\[6pt]<br \/>\n\\bigcup_{i=1}^{n} A_i &#038;= A_1 \\cup A_2 \\cup \\cdots \\cup A_n<br \/>\n\\end{align}\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \\(I = \\left\\{ 1,\\,2,\\, \\cdots ,\\, n \\right\\}\\)\uc774\ub77c\uba74 \uc704 \ub450 \uc9d1\ud569\uc740<br \/>\n\\[\\bigcap_{i=1}^{n} A_i = \\bigcap_{i\\in I} A_i ,\\,\\,\\,<br \/>\n\\bigcup_{i=1}^{n} A_i = \\bigcup_{i\\in I} A_i\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc77c\ubc18\uc801\uc73c\ub85c \\(I\\)\uac00 \uc9d1\ud569\uc774\uace0 \uc784\uc758\uc758 \\(i \\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(A_i\\)\uac00 \uc9d1\ud569\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\bigcap_{i\\in I} A_i &#038;= \\left\\{ x \\,\\vert\\, (\\forall i \\in I) (x\\in A_i) \\right\\} ,\\\\[6pt]<br \/>\n\\bigcup_{i\\in I} A_i &#038;= \\left\\{ x \\,\\vert\\, (\\exists i \\in I) (x\\in A_i) \\right\\} .<br \/>\n\\end{align}\\]<br \/>\n\uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \uad50\uc9d1\ud569\uacfc \ud569\uc9d1\ud569\uc5d0 \ub300\ud574\uc11c\ub3c4 \ub2e4\uc74c\uacfc \uac19\uc740 \ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\left( \\bigcap_{i\\in I} A_i \\right)^C &#038;= \\bigcup_{i\\in I} \\left( A_i \\right)^C ,\\\\[6pt]<br \/>\n\\left( \\bigcup_{i\\in I} A_i \\right)^C &#038;= \\bigcap_{i\\in I} \\left( A_i \\right)^C .<br \/>\n\\end{align}\\]\n<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6. (\ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569)<\/span><\/p>\n<p>\uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(A_x = [x ,\\, 5+2x]\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(I = \\left\\{ 1,\\,2,\\,3 \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\bigcup_{x\\in I} A_x &#038;= [1,\\,7] \\cup [2,\\,9] \\cup [3,\\,11] = [1,\\,11],\\\\[6pt]<br \/>\n\\bigcap_{x\\in I} A_x &#038;= [1,\\,7] \\cap [2,\\,9] \\cap [3,\\,11] = [3,\\,7]<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/h3>\n<p>\uc9d1\ud569\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uc758 \uac1c\ub150\uc744 \uc815\uc758\ud558\ub824\uba74 \uba3c\uc800 \uc9d1\ud569\uc758 \ub300\ub4f1 \uad00\uacc4\ub97c \uc815\uc758\ud574\uc57c \ud55c\ub2e4. \\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc77c\ub300\uc77c \ub300\uc751 \\(\\phi : A \\,\\to\\, B\\)\uac00 \uc874\uc7ac\ud558\uba74 \u2018\\(A\\)\uc640 \\(B\\)\ub294 <span class=\"defined\">\ub300\ub4f1\ud558\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \\(A\\,\\approx\\,B\\) \ub610\ub294 \\(A\\,\\sim\\,B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(B\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \uc911\uc5d0\uc11c \\(A\\)\uc640 \ub300\ub4f1\ud55c \uac83\uc774 \uc874\uc7ac\ud558\uba74 \\(A\\preccurlyeq B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(A\\preccurlyeq B\\)\uc774\uc9c0\ub9cc \\(A \\not\\approx B\\)\uc774\uba74 \\(A\\precnsim B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc9d1\ud569\uc758 \ud06c\uae30\uc640 \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uc911\uc694\ud55c \uc815\ub9ac\uac00 \uc788\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac<\/span> :&nbsp; \\(A\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(\\mathcal{P} (A)\\)\uac00 \\(A\\)\uc758 \uba71\uc9d1\ud569\uc774\uba74 \\(A \\precnsim \\mathcal{P} (A)\\)\uc774\ub2e4.<\/li>\n<li><span class=\"defined\">\uc288\ub8b0\ub354-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac<\/span> :&nbsp; \\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(A\\preccurlyeq B\\)\uc774\uba74\uc11c \\(B\\preccurlyeq A\\)\uc774\uba74 \\(A \\approx B\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \\(0\\) \ub610\ub294 \uc790\uc5f0\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \uc9d1\ud569\uc744 \uc720\ud55c\uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989<br \/>\n\\[\\omega = \\left\\{ 0 \\right\\} \\cup \\mathbb{N} ,\\,\\,\\, \\omega_m = \\left\\{ k \\in \\omega \\,\\vert\\, k < m \\right\\}\\]\n\uc774\ub77c\uace0 \ud558\uc790. \\(A\\)\uac00 \uc9d1\ud569\uc774\uace0\n\\[(\\exists m \\in \\omega )(A \\approx \\omega_m)\\]\n\uc774\uba74 \\(A\\)\ub97c <span class=\"defined\">\uc720\ud55c\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(A\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uace0 \\(A \\approx \\omega_m ,\\) \\(m \\in \\omega\\)\uc77c \ub54c, \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \\(\\lvert A \\rvert = m\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc720\ud55c\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc744 <span class=\"defined\">\ubb34\ud55c\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ubb34\ud55c\uc9d1\ud569 \uc911\uc5d0\uc11c \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud55c \uc9d1\ud569\uc744 <span class=\"defined\">\uac00\ubd80\ubc88\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uac00\ubd80\ubc88\uc9d1\ud569\uacfc \uc720\ud55c\uc9d1\ud569\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uac00\uc0b0\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(A\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc740 \\(A \\preccurlyeq \\mathbb{N}\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \uac00\uc0b0\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc744 <span class=\"defined\">\ube44\uac00\uc0b0\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ubb34\ud55c\uc9d1\ud569\uacfc \uac00\uc0b0\uc9d1\ud569\uc758 \uc131\uc9c8 \uc911 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uac83\ub4e4\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(A\\)\uc640 \\(B\\)\uac00 \ubaa8\ub450 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \\(A\\cup B\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc640 \\(B\\)\uac00 \ubaa8\ub450 \uac00\uc0b0\uc9d1\ud569\uc774\uba74 \\(A\\cup B\\)\ub3c4 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uace0 \\(B\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \\((A \\setminus B) \\approx A\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \ubb34\ud55c\uc9d1\ud569\uc740 \uac00\ubd80\ubc88\uc778 \ubd80\ubd84\uc9d1\ud569\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\uace0 \\(A\\subseteq B\\)\uc774\uba74 \\(B\\)\ub3c4 \ubb34\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(B\\)\uac00 \uac00\ubd80\ubc88\uc9d1\ud569\uc774\uace0 \\(A\\subseteq B\\)\uc774\uba74 \\(A\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(I\\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uace0 \uc784\uc758\uc758 \\(i\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(A_i \\)\uac00 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \\(\\bigcup_{i\\in I} A_i\\)\ub3c4 \uc720\ud55c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(I\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\uace0 \uc784\uc758\uc758 \\(i\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(A_i \\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\uba74 \\(\\bigcup_{i\\in I} A_i\\)\ub3c4 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc218\ud559\uc5d0\uc11c \uc77c\uc0c1\uc801\uc73c\ub85c \uc0ac\uc6a9\ud558\ub294 \uc9d1\ud569 \uc911\uc5d0\uc11c \\(\\mathbb{N} ,\\) \\(\\mathbb{Z},\\) \\(\\mathbb{Q}\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(\\mathbb{R},\\) \\(\\mathbb{C}\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 7. (\uac00\uc0b0\uc9d1\ud569\uc758 \uacf1\uc758 \uac00\uc0b0\uc131)<\/span><\/p>\n<p>\ubaa8\ub4e0 \uc131\ubd84\uc774 \uc790\uc5f0\uc218\uc778 \uc21c\uc11c\uc30d\ub4e4\uc758 \ubaa8\uc784<br \/>\n\\[E = \\left\\{ (p,\\,q) \\,\\vert\\, p\\in\\mathbb{N} ,\\, q\\in\\mathbb{N} \\right\\}\\]<br \/>\n\uc774 \uac00\uc0b0\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\\(E\\)\uc5d0 \uc18d\ud558\ub294 \uc21c\uc11c\uc30d \\((p,\\,q)\\)\ub294 \\(p+q\\)\uc758 \uac12\uc774 \uc791\uc740 \uac83\ubd80\ud130, \uadf8\ub9ac\uace0 \\(p+q\\)\uc758 \uac12\uc774 \uac19\uc73c\uba74 \\(p\\)\uac00 \uc791\uc740 \uac83\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[ (1,\\,1) ,\\, (1,\\,2),\\, (2,\\,1) ,\\, (1,\\,3) ,\\,(2,\\,2) ,\\,(3,\\,1) ,\\, \\cdots \\]<br \/>\n\uc774\ub807\uac8c \ub098\uc5f4\ud55c \ud6c4 \uc55e\uc5d0\uc11c\ubd80\ud130 \\(1,\\) \\(2,\\) \\(3,\\) \\(\\cdots\\)\uc640 \uac19\uc774 \uc790\uc5f0\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\uba74 \\(E\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \\(\\mathbb{N}\\)\uc758 \uc6d0\uc18c\uc5d0 \ube60\uc9d0 \uc5c6\uc774 \ud558\ub098\uc529 \ub300\uc751\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\ub294 \\(\\mathbb{N}\\)\uacfc \ub300\ub4f1\ud558\ub2e4.<\/p>\n<p>\uc774\uac83\uc744 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud560 \uc218\ub3c4 \uc788\ub2e4. \\(E\\)\uc758 \uac01 \uc6d0\uc18c \\((p,\\,q)\\)\ub97c \\(2^p \\cdot 3^q\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \\(\\phi : E \\to \\mathbb{N}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\phi\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\ub294 \\(\\mathbb{N}\\)\uc758 \ubd80\ubd84\uacfc \ub300\ub4f1\ud558\ub2e4. \ud55c\ud3b8 \uc790\uc5f0\uc218 \\(n\\)\uc744 \\((n,\\,1)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \\(\\varphi : \\mathbb{N} \\to E\\)\ub77c\uace0 \ud558\uba74 \\(\\varphi\\) \ub610\ud55c \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \\(\\mathbb{N}\\)\uc740 \\(E\\)\uc758 \ubd80\ubd84\uacfc \ub300\ub4f1\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc288\ub8b0\ub354-\ubca0\ub978\uc288\ud0c0\uc778 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc640 \\(\\mathbb{N}\\)\uc740 \ub300\ub4f1\ud558\ub2e4.<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc2e4\uc218\uacc4<\/h3>\n<p>\uc9d1\ud569\uc5d0 \uad6c\uc870\uac00 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \uadf8 \uc9d1\ud569\uacfc \uad6c\uc870\ub97c \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uacc4<\/span>(system)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c \uad6c\uc870\ub780 \uc9d1\ud569\uc758 \ud3ec\ud568 \uad00\uacc4, \ub367\uc148\uc774\ub098 \uacf1\uc148 \uac19\uc740 \uc5f0\uc0b0\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uad00\uacc4, \uc6d0\uc18c\uc758 \ub300\uc18c \uad00\uacc4, \ubd80\ubd84\uc9d1\ud569\uc744 \uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uad00\uacc4 \ub4f1\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<p>\uc2e4\uc218\uacc4\ub294 \uc0ac\uce59\uacc4\uc0b0\uc744 \uc790\uc720\ub86d\uac8c \ud560 \uc218 \uc788\ub294 \uc131\uc9c8, \uc218\uc758 \ud06c\uae30\ub97c \ube44\uad50\ud560 \uc218 \uc788\ub294 \uc131\uc9c8, \uadf9\ud55c\uc744 \ub2e4\ub8e8\uae30\uc5d0 \ucda9\ubd84\ud55c \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4. \uc774 \uc138 \uac00\uc9c0 \uc131\uc9c8\uc744 \ud558\ub098\uc529 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uccb4 \uacf5\ub9ac &#8211; \uc2e4\uc218\uacc4\uc758 \uc0ac\uce59\uacc4\uc0b0 \uc131\uc9c8<\/span><\/p>\n<p>\uc2e4\uc218\uacc4 \\(\\mathbb{R}\\)\ub294 \ub367\uc148, \uacf1\uc148\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(x+y \\in \\mathbb{R}\\)\uc774\ub2e4. (\ub367\uc148\uc5d0 \ub300\ud55c \ub2eb\ud798\uc131)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(x+y = y+x\\)\uc774\ub2e4. (\ub367\uc148\uc5d0 \ub300\ud55c \uad50\ud658\ubc95\uce59)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \ub300\ud558\uc5ec \\((x+y)+z = x+(y+z)\\)\uc774\ub2e4. (\ub367\uc148\uc5d0 \ub300\ud55c \uacb0\ud569\ubc95\uce59)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(x+0 = x\\)\uc774\ub2e4. (\ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec [\\(x+y = 0\\)\uc778 \uc2e4\uc218 \\(y\\)\uac00 \uc874\uc7ac]\ud55c\ub2e4. (\ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\cdot y \\in \\mathbb{R}\\)\uc774\ub2e4. (\uacf1\uc148\uc5d0 \ub300\ud55c \ub2eb\ud798\uc131)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(x \\cdot y = y \\cdot x\\)\uc774\ub2e4. (\uacf1\uc148\uc5d0 \ub300\ud55c \uad50\ud658\ubc95\uce59)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \ub300\ud558\uc5ec \\((x\\cdot y)\\cdot z = x\\cdot (y\\cdot z)\\)\uc774\ub2e4. (\uacf1\uc148\uc5d0 \ub300\ud55c \uacb0\ud569\ubc95\uce59)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\cdot 1 = x\\)\uc774\uba70 \\(1\\ne 0\\)\uc774\ub2e4. (\uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0)<\/li>\n<li>\\(0\\)\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec [\\(x\\cdot y = 1\\)\uc778 \uc2e4\uc218 \\(y\\)\uac00 \uc874\uc7ac]\ud55c\ub2e4. (\uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0)<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \ub300\ud558\uc5ec \\(x(y+z) = xy+xz\\)\uc774\ub2e4. (\ubd84\ubc30\ubc95\uce59)<\/li>\n<\/ul>\n<\/div>\n<p>\uc2e4\uc218\uc758 \ub367\uc148, \uacf1\uc148\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ub450 \uc218\uc758 \uacf1 \\(x\\cdot y\\)\ub294 \ubcf4\ud1b5 \uacf1\uc148 \uae30\ud638\ub97c \uc0dd\ub7b5\ud558\uc5ec \\(xy\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(0\\)\uc744 <span class=\"defined\">\ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0<\/span>, \\(1\\)\uc744 <span class=\"defined\">\uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\\(x+y =0\\)\uc77c \ub54c \\(y\\)\ub97c \\(x\\)\uc758 <span class=\"defined\">\ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70 \\(-x\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(x+(-x)=0\\)\uc774\ub2e4.<\/li>\n<li>\\(xy=1\\)\uc77c \ub54c \\(y\\)\ub97c \\(x\\)\uc758 <span class=\"defined\">\uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0<\/span> \ub610\ub294 <span class=\"defined\">\uc5ed\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uba70 \\(\\frac{1}{x}\\) \ub610\ub294 \\(x^{-1}\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\cdot \\frac{1}{x} = 1\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc2e4\uc218\uc758 \ube84\uc148, \ub098\ub217\uc148\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc77c \ub54c \\(x-y = x + (-y)\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(y \\neq 0\\)\uc77c \ub54c \\(\\frac{x}{y} = x \\cdot \\frac{1}{y}\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\ub2e4\uc74c\uc73c\ub85c \ubd80\ub4f1\ud638\uc640 \uad00\ub828\ub41c \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc21c\uc11c \uacf5\ub9ac &#8211; \uc2e4\uc218\uacc4\uc758 \uc21c\uc11c\uad00\uacc4 \uc131\uc9c8<\/span><\/p>\n<p>\uc2e4\uc218\uacc4 \\(\\mathbb{R}\\)\ub294 \ubd80\ub4f1\ud638\uc640 \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uc2e4\uc218 \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \ub300\ud558\uc5ec, \\(x < y\\)\uc774\uba74 \\(x+z < y+z\\)\uc774\ub2e4.<\/li>\n<li>\uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec, [\\(x > 0\\)\uc774\uace0 \\(y > 0\\)]\uc774\uba74 \\(xy > 0\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<p>\uc2e4\uc218\uacc4\uc758 \uc21c\uc11c\uad00\uacc4\uc640 \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(x > 0\\)\uc77c \ub54c \\(x\\)\ub97c <span class=\"defined\">\uc591\uc218<\/span>, \\(x < 0\\)\uc77c \ub54c \\(x\\)\ub97c <span class=\"defined\">\uc74c\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>[\\(x > y\\) \ub610\ub294 \\(x = y\\)]\uc77c \ub54c \\(x \\ge y\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(x > y\\)\uc640 \\(y < x\\)\ub294 \uac19\uc740 \ub73b\uc774\uba70, \\(x \\ge y\\)\uc640 \\(y \\le x\\)\ub294 \uac19\uc740 \ub73b\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(u\\)\uac00 \uc2e4\uc218\uc774\uace0 \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(x \\le u\\)\uc774\uba74 \\(u\\)\ub97c \\(E\\)\uc758 <span class=\"defined\">\uc0c1\uacc4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(\\ell\\)\uc774 \uc2e4\uc218\uc774\uace0 \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(\\ell \\le x\\)\uc774\uba74 \\(\\ell\\)\uc744 \\(E\\)\uc758 <span class=\"defined\">\ud558\uacc4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uc758 \uc0c1\uacc4\uac00 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud558\uba74 \u2018\\(E\\)\ub294 <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \\(E\\)\uc758 \ud558\uacc4\uac00 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud558\uba74 \u2018\\(E\\)\ub294 <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \\(E\\)\uac00 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \u2018\\(E\\)\ub294 <span class=\"defined\">\uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(M\\)\uacfc \\(m\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(M \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(x \\le M\\)\uc774\uba74 \\(M\\)\uc744 \\(E\\)\uc758 <span class=\"defined\">\ucd5c\ub313\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(m \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(m \\le x\\)\uc774\uba74 \\(m\\)\uc744 \\(E\\)\uc758 <span class=\"defined\">\ucd5c\uc19f\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc0c1\ud55c\uacfc \ud558\ud55c)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\\(E\\)\uac00 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uc758 \uc0c1\uacc4 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc774 \uc874\uc7ac\ud558\uba74, \uadf8\uac83\uc744 \\(E\\)\uc758 <span class=\"defined\">\ucd5c\uc18c\uc0c1\uacc4<\/span> \ub610\ub294 <span class=\"defined\">\uc0c1\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(E\\)\uc758 \uc0c1\ud55c\uc744 \\(\\operatorname{lub} E\\) \ub610\ub294 \\(\\sup E\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(E\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uc758 \ud558\uacc4 \uc911 \uac00\uc7a5 \ud070 \uac83\uc774 \uc874\uc7ac\ud558\uba74, \uadf8\uac83\uc744 \\(E\\)\uc758 <span class=\"defined\">\ucd5c\ub300\ud558\uacc4<\/span> \ub610\ub294 <span class=\"defined\">\ud558\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(E\\)\uc758 \ud558\ud55c\uc744 \\(\\operatorname{glb} E\\) \ub610\ub294 \\(\\inf E\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc9d1\ud569\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc740 \uadf8 \uc9d1\ud569\uc5d0 \uc18d\ud560 \uc218\ub3c4 \uc788\uace0 \uadf8\ub807\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \uc608\ucee8\ub300 \ub2eb\ud78c \uad6c\uac04 \\(I = [2,\\, \\pi ]\\)\uc758 \uc0c1\ud55c \\(\\pi\\)\uc640 \ud558\ud55c \\(2\\)\ub294 \\(I\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uc5f4\ub9b0 \uad6c\uac04 \\(J = (\\gamma ,\\, e)\\)\uc758 \uc0c1\ud55c \\(e\\)\uc640 \ud558\ud55c \\(\\gamma\\)\ub294 \\(J\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4. (\\(\\gamma\\)\ub294 \uc624\uc77c\ub7ec-\ub9c8\uc2a4\ucf00\ub85c\ub2c8 \uc0c1\uc218\uc774\uba70, \uadf8 \uac12\uc740 \uc57d \\(0.577\\)\uc774\ub2e4.)<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc644\ube44\uc131 \uacf5\ub9ac &#8211; \uc2e4\uc218\uacc4\uc758 \uc0c1\ud55c \uc874\uc7ac \uc131\uc9c8<\/span><\/p>\n<ul>\n<li>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc704\ub85c \uc720\uac8c\uc774\uba74 \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc544\ub798\ub85c \uc720\uac8c\uc774\uba74 \\(E\\)\uc758 \ud558\ud55c\uc774 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<p>\uc0ac\uce59\uacc4\uc0b0 \uc131\uc9c8\uacfc \uc21c\uc11c\uad00\uacc4 \uc131\uc9c8\uc740 \\(\\mathbb{R}\\)\uc640 \\(\\mathbb{Q}\\)\uac00 \uacf5\ud1b5\uc73c\ub85c \uac00\uc9c4 \uc131\uc9c8\uc774\ub2e4. \uc989 \uc774 \ub450 \uc885\ub958\uc758 \uc131\uc9c8\ub9cc\uc73c\ub85c\ub294 \\(\\mathbb{R}\\)\uc640 \\(\\mathbb{Q}\\)\ub97c \uad6c\ubd84\ud560 \uc218 \uc5c6\ub2e4. \uadf8\ub7ec\ub098 \uc0c1\ud55c \uc874\uc7ac \uc131\uc9c8\uc740 \\(\\mathbb{Q}\\)\uac00 \uac00\uc9c0\uace0 \uc788\uc9c0 \uc54a\uc740 \\(\\mathbb{R}\\)\ub9cc\uc758 \uace0\uc720\ud55c \uc131\uc9c8\uc774\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[E = \\left\\{ x \\in \\mathbb{Q} \\,\\vert\\, x^2 < 2 \\right\\}\\]\n\ub77c\uace0 \ud558\uba74 \\(E\\)\ub294 \\(\\mathbb{Q}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc720\uacc4\uc774\uc9c0\ub9cc, \\(E\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc740 \\(\\mathbb{Q}\\)\uc5d0\uc11c \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \uc138 \uac1c\uc758 \uacf5\ub9ac\ub294 \ub2e4\uc74c \uc815\uc758\uc5d0\uc11c\uc640 \uac19\uc774 \uc2e4\uc218\uacc4\ub97c \uacb0\uc815\ud558\ub294 \uc870\uac74\uc73c\ub85c \uc0ac\uc6a9\ub420 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\uc2e4\uc218\uacc4)<\/span><\/p>\n<p>\uccb4 \uacf5\ub9ac, \uc21c\uc11c \uacf5\ub9ac, \uc644\ube44\uc131 \uacf5\ub9ac\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uacc4(system)\ub294 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud558\ub294\ub370, \uadf8 \uacc4\ub97c <span class=\"defined\">\uc2e4\uc218\uacc4<\/span>(real number system)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\mathbb{R}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(F = \\left\\{ -x \\,\\vert\\, x\\in E \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(-\\alpha\\)\uac00 \\(F\\)\uc758 \ud558\ud55c\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(y \\in F\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(-y \\in E\\)\uc774\ubbc0\ub85c \\(-y \\le \\alpha\\) \uc989 \\(-\\alpha \\le y\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(-\\alpha\\)\ub294 \\(F\\)\uc758 \ud558\uacc4\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(\\beta\\)\uac00 \\(F\\)\uc758 \ud558\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x\\in E\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(-x \\in F\\)\uc774\ubbc0\ub85c \\(\\beta \\le -x\\) \uc989 \\(x\\le -\\beta\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\alpha\\)\ub294 \\(E\\)\uc758 \uc0c1\uacc4 \uc911 \ucd5c\uc19f\uac12\uc774\ubbc0\ub85c \\(\\alpha \\le -\\beta\\) \uc989 \\(\\beta \\le -\\alpha\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(-\\alpha\\)\uac00 \\(F\\)\uc758 \ud558\uacc4 \uc911 \uac00\uc7a5 \ud070 \uac12\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(-\\alpha\\)\ub294 \\(F\\)\uc758 \ud558\ud55c\uc774\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0\uc758 \ub17c\uc758 \uacfc\uc815\uc744 \uac70\uafb8\ub85c \uac70\uc2ac\ub7ec \uac00\uba74 \\(-\\alpha\\)\uac00 \\(F\\)\uc758 \ud558\ud55c\uc774\ub77c\ub294 \uac00\uc815\uc73c\ub85c\ubd80\ud130 \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc774\ub77c\ub294 \uacb0\ub860\uc744 \ub04c\uc5b4\ub0bc \uc218 \uc788\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec, \uc0c1\ud55c\uc758 \uc131\uc9c8\ub9cc \ubc1d\ud788\uba74 \ud558\ud55c\uc758 \uc131\uc9c8\uc740 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub530\ub77c\uc624\uac8c \ub41c\ub2e4.<\/p>\n<p>\uc0c1\ud55c\uc740 \u2018\uc0c1\uacc4\uc758 \ucd5c\uc19f\uac12\u2019\uc73c\ub85c \uc815\uc758\ud560 \uc218\ub3c4 \uc788\uc9c0\ub9cc \ub2e4\uc74c\uacfc \uac19\uc774 \\(\\epsilon\\) \ub17c\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\\(\\epsilon\\)\uc744 \uc774\uc6a9\ud55c \uc0c1\ud55c\uc758 \uc815\uc758)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\uacc4\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec [\\(\\alpha &#8211; \\epsilon < x \\le \\alpha\\)\uc778 \uc2e4\uc218 \\(x\\in E\\)\uac00 \uc874\uc7ac]\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\alpha &#8211; \\epsilon < x \\le \\alpha\\)\uc778 \\(x\\in E\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \\(\\alpha - \\epsilon\\)\uc740 \\(E\\)\uc758 \uc0c1\uacc4\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\alpha - \\epsilon < \\alpha\\)\uc774\ubbc0\ub85c \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\uacc4 \uc911 \ucd5c\uc19f\uac12\uc774\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\alpha - \\epsilon < x \\le \\alpha\\)\uc778 \\(x\\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\beta = \\sup E\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\alpha\\)\ub294 \\(E\\)\uc758 \uc0c1\uacc4\uc774\uc9c0\ub9cc \uc0c1\ud55c\uc740 \uc544\ub2c8\ubbc0\ub85c \\(\\beta < \\alpha\\)\uc774\ub2e4. \\(\\epsilon = \\alpha - \\beta\\)\ub77c\uace0 \ud558\uba74 \\(\\epsilon > 0\\)\uc774\uc9c0\ub9cc \\(\\alpha &#8211; \\epsilon < x \\le \\alpha\\)\uc778 \\(x\\in E\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 3. (\\(\\epsilon\\)\uc744 \uc774\uc6a9\ud55c \ud558\ud55c\uc758 \uc815\uc758)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc544\ub798\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\alpha\\)\uac00 \\(E\\)\uc758 \ud558\uacc4\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\alpha\\)\uac00 \\(E\\)\uc758 \ud558\ud55c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec [\\(\\alpha \\le x < \\alpha + \\epsilon\\)\uc778 \uc2e4\uc218 \\(x\\in E\\)\uac00 \uc874\uc7ac]\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<p>\ubbf8\uc801\ubd84\uc758 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uc0c1\ud55c\uc758 \uc131\uc9c8 \uba87 \uac00\uc9c0\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uc9d1\ud569\uc758 \uc5f0\uc0b0\uacfc \uad00\ub828\ub41c \uc0c1\ud55c\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\\(A\\)\uc640 \\(B\\)\uac00 \uc2e4\uc218 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\\((A+B) = \\left\\{ a+b \\,\\vert\\, a\\in A ,\\, b\\in B \\right\\}\\)\ub77c\uace0 \ud558\uba74 \\(\\sup (A+B) = \\sup A + \\sup B\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc6d0\uc18c\uc640 \\(B\\)\uc758 \uc6d0\uc18c\uac00 \ubaa8\ub450 \uc591\uc218\uc774\uace0 \\(AB = \\left\\{ ab \\,\\vert\\, a\\in A ,\\, b\\in B \\right\\}\\)\ub77c\uace0 \ud558\uba74 \\(\\sup(AB) = \\sup A \\cdot \\sup B \\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\alpha = \\sup A ,\\) \\(\\beta = \\sup B\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>[1] \\(x\\in (A+B)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x = a+b\\)\uc778 \\(a\\in A\\)\uc640 \\(b\\in B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(a\\le \\alpha\\)\uc774\uace0 \\(b\\le \\beta\\)\uc774\ubbc0\ub85c \\(x=a+b \\le \\alpha +\\beta\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(\\alpha + \\beta\\)\uac00 \\((A+B)\\)\uc758 \uc0c1\uacc4\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\alpha + \\beta\\)\uac00 \\((A+B)\\)\uc758 \uc0c1\ud55c\uc784\uc744 \ubcf4\uc774\uc790. \\(\\epsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\frac{\\epsilon}{2} > 0\\)\uc774\ubbc0\ub85c, \\(\\alpha &#8211; \\frac{\\epsilon}{2} < a \\le \\alpha\\)\uc778 \\(a\\in A\\)\uac00 \uc874\uc7ac\ud558\uace0, \\(\\beta - \\frac{\\epsilon}{2} < b \\le \\beta\\)\uc778 \\(b\\in B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c\n\\[ (\\alpha + \\beta) - \\epsilon < a+b \\le \\alpha + \\beta \\]\n\uc774\uace0 \\(a+b \\in (A+B)\\)\uc774\ubbc0\ub85c \\(\\alpha + \\beta\\)\ub294 \\((A+B)\\)\uc758 \uc0c1\ud55c\uc774\ub2e4.<\/p>\n<p>[2] \\(x\\in AB\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x = ab\\)\uc778 \\(a\\in A\\)\uc640 \\(b\\in B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(a\\le \\alpha\\)\uc774\uace0 \\(b\\in \\beta\\)\uc774\uace0 \\(a\\)\uc640 \\(b\\)\uac00 \uc591\uc218\uc774\ubbc0\ub85c \\(x=ab \\le \\alpha \\beta\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(\\alpha \\beta\\)\uac00 \\(AB\\)\uc758 \uc0c1\uacc4\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\alpha \\beta\\)\uac00 \\(AB\\)\uc758 \uc0c1\ud55c\uc784\uc744 \ubcf4\uc774\uc790. \\(\\epsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[\\eta = \\min\\left\\{ \\frac{\\epsilon}{\\alpha + \\beta} ,\\, \\alpha ,\\, \\beta \\right\\}\\]\ub77c\uace0 \ud558\uc790. \\(\\alpha\\)\uc640 \\(\\beta\\)\uac00 \ubaa8\ub450 \uc591\uc218\uc774\ubbc0\ub85c \\(\\eta > 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\alpha &#8211; \\eta < a \\le \\alpha\\)\uc778 \\(a\\in A\\)\uac00 \uc874\uc7ac\ud558\uace0, \\(\\beta - \\eta < b \\le \\beta\\)\uc778 \\(b\\in B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c\n\\[\\begin{align}\n\\alpha\\beta - \\epsilon &#038;\\le \\alpha\\beta -\\eta (\\alpha + \\beta) \\\\[8pt]\n&#038;< \\alpha\\beta -\\eta (\\alpha + \\beta) + \\eta ^2 \\\\[8pt]\n&#038;= (\\alpha - \\eta)(\\beta - \\eta) < ab \\le \\alpha\\beta \n\\end{align}\\]\n\uc774\uace0 \\(ab \\in AB\\)\uc774\ubbc0\ub85c \\(\\alpha\\beta\\)\ub294 \\(AB\\)\uc758 \uc0c1\ud55c\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(f\\)\uac00 \ud568\uc218\uc774\uace0 \\(E\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc9d1\ud569 \\(\\left\\{ f(x) \\,\\vert\\, x\\in E\\right\\}\\)\uac00 \uc704\ub85c \uc720\uacc4\uc774\uba74 \u2018\\(f\\)\ub294 \\(E\\)\uc5d0\uc11c <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uba70, \ub9cc\uc57d \\(\\left\\{ f(x) \\,\\vert\\, x\\in E\\right\\}\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \u2018\\(f\\)\ub294 \\(E\\)\uc5d0\uc11c <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \\(\\left\\{ f(x) \\,\\vert\\, x\\in E\\right\\}\\)\uac00 \uc720\uac8c\uc774\uba74 \u2018\\(f\\)\ub294 \\(E\\)\uc5d0\uc11c <span class=\"defined\">\uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \ud568\uc218\uc774\uace0 \\(E\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \ubd80\ubd84\uc9d1\ud569\uc77c \ub54c<br \/>\n\\[\\sup_{x\\in E}f(x) = \\sup \\left\\{ f(x) \\,\\vert\\, x\\in E \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 5. (\ud568\uc218\uc758 \uc5f0\uc0b0\uacfc \uad00\ub828\ub41c \uc0c1\ud55c\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\\(f\\)\uc640 \\(g\\)\uac00 \ud568\uc218\uc774\uace0 \\(E\\)\uac00 \\(f\\)\uc640 \\(g\\)\uc758 \uacf5\ud1b5\uc815\uc758\uc5ed\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uc640 \\(g\\)\uac00 \\(E\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\uba74<br \/>\n\\[\\sup_{x\\in E} (f(x)+g(x)) \\le \\sup_{x\\in E} f(x) + \\sup_{x\\in E}g(x)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(E\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uc744 \\(\\alpha\\)\ub77c\uace0 \ud558\uace0, \\(g\\)\uc758 \uc0c1\ud55c\uc744 \\(\\beta\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le \\alpha\\)\uc774\uace0 \\(g(x) \\le \\beta\\)\uc774\ubbc0\ub85c \\(f(x)+g(x) \\le \\alpha + \\beta\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\alpha + \\beta\\)\ub294 \uc9d1\ud569 \\(\\left\\{ f(x)+g(x) \\,\\vert\\, x\\in E \\right\\}\\)\uc758 \uc0c1\uacc4\uc774\ub2e4. \uc0c1\ud55c\uc740 \uc0c1\uacc4 \uc911 \uac00\uc7a5 \uc791\uc740 \uac12\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n\\sup_{x\\in E} (f(x)+g(x)) &#038;= \\sup \\left\\{ f(x)+g(x) \\,\\vert\\, x\\in E \\right\\} \\\\[0pt]<br \/>\n&#038;\\le \\alpha + \\beta = \\sup_{x\\in E} f(x) + \\sup_{x\\in E}g(x)<br \/>\n\\end{align}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc790\uc5f0\uc218, \uc815\uc218, \uc720\ub9ac\uc218<\/h3>\n<p>\uc790\uc5f0\uc218, \uc815\uc218, \uc720\ub9ac\uc218 \uc9d1\ud569\uc744 \uba85\ud655\ud558\uac8c \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uac04\ub2e8\ud788 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 3. (\uc218\uccb4\uacc4)<\/span><\/p>\n<ol class=\"bracket\">\n<li>\\(A\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \ub450 \uc870\uac74<br \/>\n\\[1\\in A ,\\,\\quad (n \\in A \\,\\rightarrow\\, (n+1) \\in A) \\]<br \/>\n\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(A\\)\ub97c <span class=\"defined\">\uadc0\ub0a9\uc801 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uadc0\ub0a9\uc801 \uc9d1\ud569 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc744 <span class=\"defined\">\uc790\uc5f0\uc218 \uc9d1\ud569<\/span> \\(\\mathbb{N}\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\uc815\uc218 \uc9d1\ud569<\/span>\uc744\\[\\mathbb{Z} = \\left\\{ n \\,\\vert\\, n \\in \\mathbb{N} \\text{ or } n =0 \\text{ or } -n\\in\\mathbb{N} \\right\\} \\]\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\uc720\ub9ac\uc218 \uc9d1\ud569<\/span>\uc744\\[\\mathbb{Q} = \\left\\{ \\frac{p}{q} \\, \\bigg| \\,\\,p \\in \\mathbb{Z} ,\\, q \\in \\mathbb{Z} ,\\, q \\ne 0  \\right\\} \\]\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\uc720\ub9ac\uc218\uac00 \uc544\ub2cc \uc2e4\uc218\ub97c <span class=\"defined\">\ubb34\ub9ac\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc790\uc5f0\uc218 \uc9d1\ud569\uc740 \ub367\uc148\uacfc \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\uc73c\uba70, \uc815\uc218 \uc9d1\ud569\uc740 \ub367\uc148, \ube84\uc148, \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\ub2e4. \uc720\ub9ac\uc218 \uc9d1\ud569\uc740 \ub367\uc148, \ube84\uc148, \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\uc73c\uba70, \\(0\\)\uc73c\ub85c \ub098\ub204\ub294 \uacbd\uc6b0\ub97c \uc81c\uc678\ud558\uba74 \uc720\ub9ac\uc218 \uc9d1\ud569 \ub0b4\uc5d0\uc11c \ub098\ub217\uc148\ub3c4 \uc790\uc720\ub86d\uac8c \ud560 \uc218 \uc788\ub2e4. \uc774\ub4e4 \uc9d1\ud569\uc740 \uc2e4\uc218 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \uc2e4\uc218\uacc4\uc5d0\uc11c \uc131\ub9bd\ud558\ub294 \ubd80\ub4f1\ud638\uc640 \uad00\ub828\ub41c \ubc95\uce59\uc740 \uc774\ub4e4 \uc9d1\ud569\uc5d0\uc11c\ub3c4 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4. \uc774\uc640 \uac19\uc774 \ub367\uc148, \uacf1\uc148, \ubd80\ub4f1\ud638\uc640 \uad00\ub828\ub41c \uad6c\uc870\ub97c \uac00\uc9c0\uace0 \uc788\uae30 \ub54c\ubb38\uc5d0 \uc774\ub4e4 \uc9d1\ud569\uc744 \uac01\uac01 <span class=\"defined\">\uc790\uc5f0\uc218\uacc4<\/span>, <span class=\"defined\">\uc815\uc218\uacc4<\/span>, <span class=\"defined\">\uc720\ub9ac\uc218\uacc4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 6. (\uc218\ud559\uc801 \uadc0\ub0a9\ubc95)<\/span><\/p>\n<p>\\(p(n)\\)\uc774 \\(\\mathbb{N}\\)\uc744 \uc815\uc758\uc5ed\uc73c\ub85c \ud558\ub294 \uba85\uc81c\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc989 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc8fc\uc5b4\uc9c8 \ub54c\ub9c8\ub2e4 \\(p(n)\\)\uc740 \ucc38 \ub610\ub294 \uac70\uc9d3\uc778 \uac12\uc744 \uac16\ub294 \ud568\uc218\uc774\ub2e4. \ub9cc\uc57d \\(p(n)\\)\uc774 \ub450 \uc870\uac74<\/p>\n<ol class=\"bracket\">\n<li>\\(p(1)\\)\uc774 \ucc38\uc774\ub2e4,<\/li>\n<li>\\(p(k)\\)\uac00 \ucc38\uc77c \ub54c\ub9c8\ub2e4 \\(p(k+1)\\)\ub3c4 \ucc38\uc774\ub2e4<\/li>\n<\/ol>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(p(n)\\)\uc740 \ucc38\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(p\\)\uc758 \uc9c4\ub9ac\uc9d1\ud569\uc744 \\(T = \\left\\{ n \\in \\mathbb{N} \\,\\vert\\, p(n) \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uc989 \\(T\\)\ub294 \\(p(n)\\)\uc774 \ucc38\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \\(n\\)\uc744 \ubaa8\ub450 \ubaa8\uc740 \uc9d1\ud569\uc774\ub2e4. \uba85\ubc31\ud788 \\(T\\subseteq\\mathbb{N}\\)\uc774\ub2e4. \uc774\uc81c \uc774 \ud3ec\ud568\uad00\uacc4\uac00 \ubc18\ub300\ub85c \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\uba3c\uc800 \\(p(1)\\)\uc774 \ucc38\uc774\ubbc0\ub85c \\(1\\in T\\)\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(k\\in T\\)\uc774\uba74 \\(p(k)\\)\uac00 \ucc38\uc774\ubbc0\ub85c [2]\uc5d0 \uc758\ud558\uc5ec \\(p(k+1)\\)\ub3c4 \ucc38\uc774\ub2e4. \ub530\ub77c\uc11c \\(k+1 \\in T\\)\uc774\ub2e4. \uc774\ub85c\uc368 \\(T\\)\ub294 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc784\uc774 \ubc1d\ud600\uc84c\ub2e4. \\(\\mathbb{N}\\)\uc740 \uadc0\ub0a9\uc801 \uc9d1\ud569 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc774\ubbc0\ub85c \\(\\mathbb{N} \\subseteq T\\)\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \\(T = \\mathbb{N}\\)\uc784\uc774 \ubc1d\ud600\uc84c\ub2e4. \uadf8\ub7f0\ub370 \\(T\\)\ub294 \\(p(n)\\)\uc774 \ucc38\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \\(n\\)\uc744 \ubaa8\uc740 \uc9d1\ud569\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(p(n)\\)\uc740 \ucc38\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 7. (\uc720\ud55c\uc9d1\ud569\uc758 \ucd5c\ub300 \ucd5c\uc18c)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc720\ud55c\uc9d1\ud569\uc774\uba74 \\(E\\)\ub294 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790. \\(E\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \\(n\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(n=1\\)\uc778 \uacbd\uc6b0, \uc989 \\(E\\)\uac00 \uc6d0\uc18c\ub97c \ub2e8 \ud558\ub098 \uac00\uc9c4 \uc9d1\ud569\uc774\ub77c\uba74, \uadf8 \uc6d0\uc18c\uac00 \\(E\\)\uc758 \ucd5c\ub313\uac12\uc778 \ub3d9\uc2dc\uc5d0 \ucd5c\uc19f\uac12\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(n=k\\)\uc778 \uacbd\uc6b0\ub97c \uac00\uc815\ud558\uc790. \uc989 \\(k\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\uc740 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(E\\)\uac00 \\((k+1)\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(E\\)\uc758 \uc6d0\uc18c \ud558\ub098\ub97c \ud0dd\ud558\uc5ec \\(x\\)\ub77c\uace0 \ud558\uace0, \\(E_x = E \\setminus \\left\\{ x \\right\\}\\)\ub77c\uace0 \ud558\uc790. \\(E_x\\)\ub294 \\(k\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\uc774\ubbc0\ub85c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(E_x\\)\uc758 \ucd5c\ub313\uac12\uc744 \\(M,\\) \ucd5c\uc19f\uac12\uc744 \\(m\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(M\\)\uacfc \\(x\\) \uc911 \ub354 \ud070 \uac83\uc774 \\(E\\)\uc758 \ucd5c\ub313\uac12\uc774 \ub418\uba70, \\(m\\)\uacfc \\(x\\) \uc911 \ub354 \uc791\uc740 \uac83\uc774 \\(E\\)\uc758 \ucd5c\uc19f\uac12\uc774 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\((k+1)\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\ub3c4 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<p>\ub530\ub77c\uc11c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(n\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \uc9d1\ud569\uc740 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 8. (\uc790\uc5f0\uc218 \uc9d1\ud569\uc758 \ube44\uc720\uacc4\uc131)<\/span><\/p>\n<p>\uc790\uc5f0\uc218 \uc9d1\ud569 \\(\\mathbb{N}\\)\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\mathbb{N}\\)\uc774 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131\uc5d0 \uc758\ud558\uc5ec \\(\\mathbb{N}\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc0c1\ud55c\uc744 \\(\\alpha\\)\ub77c\uace0 \ud558\uc790. \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(\\alpha -1 < n \\le \\alpha\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc790\uc5f0\uc218 \uc9d1\ud569\uc740 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc774\ubbc0\ub85c \\(n+1\\) \ub610\ud55c \uc790\uc5f0\uc218\uc774\ub2e4. \uadf8\ub7ec\uba74 \\(\\alpha < n+1\\)\uc774 \ub418\ub294\ub370, \uc774\uac83\uc740 \\(\\alpha\\)\uac00 \\(\\mathbb{N}\\)\uc758 \uc0c1\uacc4 \uc911 \ud558\ub098\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 9. (\uc790\uc5f0\uc218\uacc4\uc758 \uc815\ub82c\uc131)<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{N}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba74 \\(E\\)\ub294 \ucd5c\uc18c\uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ubbc0\ub85c \uc6d0\uc18c\ub97c \ud558\ub098 \ud0dd\ud558\uc5ec \\(k\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[E_k = \\left\\{ n \\in E \\,\\vert\\, n\\le k \\right\\}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(E_k\\)\ub294 \\(\\left\\{1,\\,2,\\,3,\\,\\cdots,\\,k\\right\\}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \uc720\ud55c\uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \\(k\\in E_k\\)\uc774\ubbc0\ub85c \\(E_k \\ne \\varnothing\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E_k\\)\ub294 \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \uadf8 \ucd5c\uc19f\uac12\uc744 \\(m\\)\uc774\ub77c\uace0 \ud558\uc790. \\(m\\in E_k\\)\uc774\ubbc0\ub85c \\(m\\in E\\)\uc774\ub2e4. \ub610\ud55c \\(E\\)\uc758 \uc6d0\uc18c \\(n\\) \uc911\uc5d0\uc11c \\(E_k\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uac83\uc740 \\(k\\)\ubcf4\ub2e4 \ud06c\ubbc0\ub85c \\(m\\le k < n\\)\uc774\uace0, \\(E_k\\)\uc5d0 \uc18d\ud558\ub294 \uac83\uc740 \ub2f9\uc5f0\ud788 \\(m\\le n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \\(m\\)\uc740 \\(E\\)\uc758 \ucd5c\uc18c\uc6d0\uc18c\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 10. (\uc720\ub9ac\uc218 \uc9d1\ud569\uc758 \uc870\ubc00\uc131)<\/span><\/p>\n<p>\\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(x < y\\)\uc774\uba74, \\(x < r < y\\)\uc778 \uc720\ub9ac\uc218 \\(r\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(x \\ge 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(\\mathbb{N}\\)\uc774 \uc704\ub85c \uc720\uac8c\uac00 \uc544\ub2c8\ubbc0\ub85c \\(\\frac{1}{y-x} < q\\)\uc778 \uc790\uc5f0\uc218 \\(q\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(\\frac{1}{q} < y-x\\)\uc774\ub2e4. \uc774\uc81c \\(yq\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218 \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac83\uc744 \\(p\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[ \\frac{p}{q} \\ge y \\]\n\uc774\uace0\n\\[\\frac{p-1}{q} < y\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370\n\\[x < y - \\frac{1}{q} \\le \\frac{p}{q} - \\frac{1}{q} = \\frac{p-1}{q} < y\\]\n\uc774\ubbc0\ub85c \\(r = \\frac{p-1}{q}\\)\uc740 \\(x\\)\uc640 \\(y\\) \uc0ac\uc774\uc5d0 \uc788\ub294 \uc720\ub9ac\uc218\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(x < 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(\\mathbb{N}\\)\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \\(n \\ge -x\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(0 \\le x+n < y+n\\)\uc774\ubbc0\ub85c, \uc99d\uba85 \uc55e\ubd80\ubd84\uc758 \ub17c\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec, \\(x+n < s < y+n\\)\uc778 \uc720\ub9ac\uc218 \\(s\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(r = s-n\\)\uc774\ub77c\uace0 \ud558\uba74 \\(r\\)\ub294 \\(x\\)\uc640 \\(y\\) \uc0ac\uc774\uc5d0 \uc788\ub294 \uc720\ub9ac\uc218\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 11. (\ubb34\ub9ac\uc218 \uc9d1\ud569\uc758 \uc870\ubc00\uc131)<\/span><\/p>\n<p>\\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(x < y\\)\uc774\uba74, \\(x < s < y\\)\uc778 \ubb34\ub9ac\uc218 \\(s\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\((\\sqrt{2} -1)x < (\\sqrt{2}-1)y\\)\uc774\ubbc0\ub85c \uc720\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec\n\\[(\\sqrt{2} -1)x < r < (\\sqrt{2}-1)y\\]\n\uc778 \uc720\ub9ac\uc218 \\(r\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc704 \uc2dd\uc758 \uac01 \ubcc0\uc5d0 \\((\\sqrt{2}+1)\\)\uc744 \uacf1\ud558\uba74\n\\[x < (\\sqrt{2}+1)r < y\\]\n\uc774\ub2e4. \\(s = (\\sqrt{2}+1)r\\)\ub77c\uace0 \ud558\uba74 \\(s\\)\ub294 \\(x\\)\uc640 \\(y\\) \uc0ac\uc774\uc5d0 \uc788\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 .<\/span><\/p>\n\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>...\n\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubbf8\uc801\ubd84\ud559\uc744 \uacf5\ubd80\ud558\uae30 \uc704\ud574 \ud544\uc694\ud55c \uae30\ucd08 \uac1c\ub150\uc778 \uc218\ud559\uc758 \ub17c\ub9ac, \uc9d1\ud569\uc758 \uc131\uc9c8, \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8\uc744 \uac04\ub7b5\ud558\uac8c \uc0b4\ud3b4\ubcf8\ub2e4. \uc218\ud559\uc758 \ub17c\ub9ac \uba3c\uc800 \ub17c\ub9ac\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(p\\)\uc640 \\(q\\)\uac00 \uc218\ud559\uc801 \ubb38\uc7a5\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \ub17c\ub9ac\uacf1 :&nbsp; \u2018\\(p\\) \uadf8\ub9ac\uace0 \\(q\\)\u2019\ub97c \\((p \\,\\wedge\\, q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((p \\,\\wedge\\, q)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \\(p\\)\uc640 \\(q\\)\uac00 \ubaa8\ub450 \ucc38\uc774\ub77c\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub17c\ub9ac\ud569:&nbsp; \u2018\\(p\\) \ub610\ub294 \\(q\\)\u2019\ub97c \\((p\\,\\vee\\,q)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\((p\\,\\vee\\,q)\\)\uac00 \ucc38\uc774\ub77c\ub294 \uac83\uc740 \\(p\\)\uc640 \\(q\\) \uc911 \ud558\ub098&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[115,109,105,111,107,118,101,120,113,103,117,108,100,104,106,116,119,112,114,121,122,102,110],"class_list":["post-2835","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-completeness","tag-countable-set","tag-de-morgan","tag-field-axiom","tag-infinite-set","tag-least-upper-bound","tag-logic","tag-mathematical-induction","tag-order-axiom","tag-set","tag-supremum","tag-108","tag-100","tag-104","tag-106","tag-116","tag-119","tag-112","tag-114","tag-121","tag-122","tag-102","tag-110"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2835","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\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2835"}],"version-history":[{"count":111,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2835\/revisions"}],"predecessor-version":[{"id":3304,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2835\/revisions\/3304"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2835"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2835"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2835"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}