{"id":6972,"date":"2021-07-23T23:17:48","date_gmt":"2021-07-23T14:17:48","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6972"},"modified":"2021-08-21T18:30:30","modified_gmt":"2021-08-21T09:30:30","slug":"algebra-of-sets","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/algebra-of-sets\/","title":{"rendered":"\uc9d1\ud569\uc758 \uc5f0\uc0b0"},"content":{"rendered":"<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \ud45c\uae30\ubc95\uacfc \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc5ec\uae30\uc11c\ub294 \uc9d1\ud569\uc758 \uc9c1\uad00\uc801 \uc815\uc758\ub97c \ubc1b\uc544\ub4e4\uc778\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc9d1\ud569\uacfc \uc6d0\uc18c<\/h2>\n<ul>\n<li>\uc218\ud559\uc801 \uac1c\uccb4 \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud560 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(a\\in A\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc218\ud559\uc801 \uac1c\uccb4 \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(a\\notin A\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\ub450 \uc9d1\ud569 \\(A,\\) \\(B\\)\uac00 \uac19\uc740 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(A=B\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(A\\)\uc5d0 \uc18d\ud55c \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\(B\\)\uc5d0 \uc18d\ud560 \ub54c, \u201c\\(A\\)\uac00 \\(B\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc9d1\ud569<\/span>\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(A\\subseteq B\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. (\ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \uc774\uac83\uc744 \\(A\\subset B\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.)<\/li>\n<li>\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(A\\subseteq B\\)\uc774\uc9c0\ub9cc \\(A\\ne B\\)\uc77c \ub54c, \u201c\\(A\\)\uac00 \\(B\\)\uc758 <span class=\"defined\">\uc9c4\ubd80\ubd84\uc9d1\ud569<\/span>\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(A\\subsetneq B\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc9d1\ud569 \\(A\\)\uac00 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c \\(x_1 ,\\) \\(x_2 ,\\) \\(\\cdots ,\\) \\(x_n \\)\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c<br \/>\n\\[A = \\left\\{ x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc9d1\ud569 \\(B\\)\uac00 \\(U\\)\uc758 \uc6d0\uc18c \uc911 \uc131\uc9c8 \\(p(x)\\)\ub97c \uac16\ub294 \uac83\ub4e4\ub9cc\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c<br \/>\n\\[B = \\left\\{ x\\in U \\,\\vert\\, p(x) \\right\\}\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc9d1\ud569\uc758 \uc5f0\uc0b0<\/h2>\n<p>\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\ud569\uc9d1\ud569<\/span>(union)\uc744<br \/>\n\\[A\\cup B = \\left\\{ x \\,\\vert\\, x\\in A \\,\\,\\text{or}\\,\\, x\\in B \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\uad50\uc9d1\ud569<\/span>(intersection)\uc744<br \/>\n\\[A\\cap B = \\left\\{ x \\,\\vert\\, x\\in A \\,\\,\\text{and}\\,\\, x\\in B \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<li>\\(A\\)\uc5d0\uc11c \\(B\\)\ub97c \ube80 <span class=\"defined\">\ucc28\uc9d1\ud569<\/span>(difference)\uc744<br \/>\n\\[A\\setminus B = \\left\\{ x \\,\\vert\\, x\\in A \\,\\,\\text{and}\\,\\, x\\notin B \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\uc9d1\ud569 \\(U\\)\ub97c \uc804\uccb4\uc9d1\ud569\uacfc \uac19\uc774 \uace0\uc815\ud574 \ub450\uace0, \\(U\\)\uc758 \uc6d0\uc18c\ub9cc\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc744 \ub17c\ud560 \ub54c\uac00 \uc788\ub2e4. \uc774\ub54c \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc740 \uc6d0\uc18c\ub4e4\ub9cc \ubaa8\uc740 \uc9d1\ud569\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc5ec\uc9d1\ud569<\/span>(complement)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A^c\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(U\\)\uac00 \uc804\uccb4\uc9d1\ud569\uc77c \ub54c<br \/>\n\\(A^c = U \\setminus A\\)<br \/>\n\uc774\ub2e4.<\/p>\n<p>\ub450 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc740 \uc784\uc758\uc758 \uac1c\uc218\uc758 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uacfc \uad50\uc9d1\ud569\uc73c\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4. \\(A_1 ,\\) \\(A_2 ,\\) \\(\\cdots ,\\) \\(A_n\\)\uc774 \uc9d1\ud569\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\bigcup_{i=1}^{n} A_i &#038;= A_1 \\cup A_2 \\cup \\cdots \\cup A_n ,\\\\[6pt]<br \/>\n\\bigcap_{i=1}^{n} A_i &#038;= A_1 \\cap A_2 \\cap \\cdots \\cap A_n .<br \/>\n\\end{align}\\]<br \/>\n\ub610\ud55c \\(\\left\\{ A_i \\,\\vert\\, i\\in I\\right\\}\\)\uac00 \uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\bigcup_{i\\in I} A_i &#038;= \\left\\{ x\\,\\vert\\, x\\in A_i \\text{ for at least one } i \\in I \\right\\} ,\\\\[6pt]<br \/>\n\\bigcap_{i\\in I} A_i &#038;= \\left\\{ x\\,\\vert\\, x\\in A_i \\text{ for all } i \\in I \\right\\} .<br \/>\n\\end{align}\\]<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ub370\uce74\ub974\ud2b8 \uacf1<\/h2>\n<p>\\(A\\)\uc640 \\(B\\)\uac00 \uc9d1\ud569\uc77c \ub54c, \\(A\\)\uc640 \\(B\\)\uc758 <span class=\"defined\">\ub370\uce74\ub974\ud2b8 \uacf1<\/span>(Cartesian product)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[A \\times B = \\left\\{ (x,\\,y) \\,\\vert\\, x\\in A \\text{ and } y\\in B \\right\\}.\\]<br \/>\n\ub610\ud55c \\(A_1 ,\\) \\(A_2 ,\\) \\(\\cdots ,\\) \\(A_n\\)\uc774 \uc9d1\ud569\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\prod_{i=1}^{n} A_i = A_1 \\times A_2 \\times \\cdots \\times A_n<br \/>\n= \\left\\{ \\left( x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right) \\,\\vert\\, x_i \\in A_i \\text{ for all } i\\in I \\right\\}.\\]<br \/>\n\ub9cc\uc57d \\(A_1 = A_2 = \\cdots = A_n = A\\)\uc774\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[A^n = \\prod_{i=1}^n A = \\left\\{ \\left( x_1 ,\\, x_2 ,\\,\\cdots ,\\, x_n \\right) \\,\\vert\\, x_i \\in A \\text{ for all } i \\in I \\right\\}.\\]<\/p>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\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<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/functions\">\ud568\uc218<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc774 \uc808\uc5d0\uc11c\ub294 \uc9d1\ud569\uc758 \ud45c\uae30\ubc95\uacfc \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc5ec\uae30\uc11c\ub294 \uc9d1\ud569\uc758 \uc9c1\uad00\uc801 \uc815\uc758\ub97c \ubc1b\uc544\ub4e4\uc778\ub2e4. \uc9d1\ud569\uacfc \uc6d0\uc18c \uc218\ud559\uc801 \uac1c\uccb4 \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud560 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(a\\in A\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc218\ud559\uc801 \uac1c\uccb4 \\(a\\)\uac00 \uc9d1\ud569 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(a\\notin A\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ub450 \uc9d1\ud569 \\(A,\\) \\(B\\)\uac00 \uac19\uc740 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc744 \ub54c, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(A=B\\)\uc640 \uac19\uc774&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":11,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6972","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6972","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=6972"}],"version-history":[{"count":14,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6972\/revisions"}],"predecessor-version":[{"id":7229,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6972\/revisions\/7229"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}