{"id":7216,"date":"2011-11-27T22:42:22","date_gmt":"2011-11-27T22:42:22","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7216"},"modified":"2021-12-28T12:46:02","modified_gmt":"2021-12-28T12:46:02","slug":"escreva-todos-os-subconjuntos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7216","title":{"rendered":"Escreva todos os subconjuntos"},"content":{"rendered":"<p><ul id='GTTabs_ul_7216' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7216' class='GTTabs_curr'><a  id=\"7216_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7216' ><a  id=\"7216_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7216'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Escreva todos os subconjuntos do conjunto $A=\\left\\{ d,e,f \\right\\}$.<br \/>\n(N\u00e3o esque\u00e7a que o conjunto vazio e o conjunto A s\u00e3o subconjuntos de A.)<\/li>\n<li>Transcreva e complete o quadro seguinte e compare-o com as 5 primeiras linhas do Tri\u00e2ngulo de Pascal.<br \/>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<caption>N\u00famero de subconjuntos de B com:<\/caption>\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">Subcomjuntos de B<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">0 elementos<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">1 elementos<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">2 elementos<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">3 elementos<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">4 elementos<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">Total<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$\\left\\{ {} \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$1={{2}^{0}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">\u00a0$\\left\\{ a \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$2={{2}^{1}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">\u00a0$\\left\\{ a,b \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$4={{2}^{2}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">\u00a0$\\left\\{ a,b,c \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$&#8230;=&#8230;$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">\u00a0$\\left\\{ a,b,c,d \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #ffa07a 1px solid;\">$&#8230;=&#8230;$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Mostre que ${}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+{}^{n}{{C}_{2}}+&#8230;+{}^{n}{{C}_{n}}={{2}^{n}}$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7216' onClick='GTTabs_show(1,7216)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7216'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Subconjuntos do conjunto $A=\\left\\{ d,e,f \\right\\}$, com:\n<p>&#8211; zero elementos: $\\left\\{ {} \\right\\}$<\/p>\n<p>&#8211; um elemtento: $\\left\\{ d \\right\\}$, $\\left\\{ e \\right\\}$ e $\\left\\{ f \\right\\}$<\/p>\n<p>&#8211; dois elementos: $\\left\\{ d,e \\right\\}$, $\\left\\{ d,f \\right\\}$ e $\\left\\{ e,f \\right\\}$<\/p>\n<p>&#8211; tr\u00eas elementos: $\\left\\{ d,e,f \\right\\}$<\/p>\n<\/li>\n<li>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<caption>N\u00famero de subconjuntos de B com:<\/caption>\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">Subcomjuntos de B<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">0 elementos<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">1 elementos<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">2 elementos<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">3 elementos<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">4 elementos<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">Total<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">$\\left\\{ {} \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">$1={{2}^{0}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">\u00a0$\\left\\{ a \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">$2={{2}^{1}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">\u00a0$\\left\\{ a,b \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">2<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">$4={{2}^{2}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">\u00a0$\\left\\{ a,b,c \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">3<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">3<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">&#8211;<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">$8={{2}^{3}}$<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\">\u00a0$\\left\\{ a,b,c,d \\right\\}$<\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">4<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">6<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">4<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"text-align: center; border: #ff7f50 1px solid;\"><span style=\"color: #0000ff;\">$16={{2}^{4}}$<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0$$\\begin{matrix}<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; 1 &amp; {} &amp; {} &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; {} &amp; 1 &amp; {} &amp; 1 &amp; {} &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; 1 &amp; {} &amp; 2 &amp; {} &amp; 1 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; 1 &amp; {} &amp; 3 &amp; {} &amp; 3 &amp; {} &amp; 1 &amp; {}\u00a0 \\\\<br \/>\n1 &amp; {} &amp; 4 &amp; {} &amp; 6 &amp; {} &amp; 4 &amp; {} &amp; 1\u00a0 \\\\<br \/>\n\\end{matrix}$$<\/p>\n<\/li>\n<li>Por \u00faltimo:<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\n{}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+{}^{n}{{C}_{2}}+&#8230;+{}^{n}{{C}_{n}} &amp; = &amp; \\sum\\limits_{k=0}^{n}{{}^{n}{{C}_{k}}\\times {{1}^{n-k}}\\times {{1}^{k}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{(1+1)}^{n}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{2}^{n}}\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7216' onClick='GTTabs_show(0,7216)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Escreva todos os subconjuntos do conjunto $A=\\left\\{ d,e,f \\right\\}$. (N\u00e3o esque\u00e7a que o conjunto vazio e o conjunto A s\u00e3o subconjuntos de A.) Transcreva e complete o quadro seguinte e compare-o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19177,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[226,97,227],"tags":[427,257,256],"series":[],"class_list":["post-7216","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-binomio-de-newton","tag-triangulo-de-pascal"],"views":1814,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat68.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7216","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7216"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7216\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7216"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7216"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7216"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}