{"id":60301,"date":"2022-10-25T10:30:00","date_gmt":"2022-10-25T13:30:00","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=60301"},"modified":"2022-11-06T10:35:16","modified_gmt":"2022-11-06T13:35:16","slug":"relacao-binaria","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/","title":{"rendered":"O que \u00e9 uma Rela\u00e7\u00e3o Bin\u00e1ria na Matem\u00e1tica?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Se voc\u00ea tem o sonho de ingressar em algum dos seletivos das <span id=\"urn:enhancement-fbeb157d-6545-41fa-880f-f76e2c8a0eaa\" class=\"textannotation disambiguated wl-creative-work wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110246\/entity\/forcas_armadas\">For\u00e7as Armadas<\/span>, sabe que a mat\u00e9ria de <strong>Matem\u00e1tica \u00e9 uma das mais cobradas nas avalia\u00e7\u00f5es. <\/strong>Dentro dos conhecimentos b\u00e1sicos para o tema de fun\u00e7\u00f5es, j\u00e1 trouxemos para voc\u00ea os resumos de<a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\" target=\"_blank\" > par ordenado e produto cartesiano<\/a>. Agora, <strong>entenda o que \u00e9 Rela\u00e7\u00e3o Bin\u00e1ria e seus tipos!&nbsp;<\/strong><\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado\t<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Qual-a-definicao-de-relacao-binaria\" >Qual a defini\u00e7\u00e3o de rela\u00e7\u00e3o bin\u00e1ria?&nbsp;<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Inscreva-se-em-nossa-newsletter\" >Inscreva-se em nossa newsletter!<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Dominio-e-Imagem\" >Dom\u00ednio e Imagem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Tipos-de-Relacao-Binaria\" >Tipos de Rela\u00e7\u00e3o Bin\u00e1ria&nbsp;<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Reflexiva\" >Reflexiva<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Simetrica\" >Sim\u00e9trica<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Antissimetrica\" >Antissim\u00e9trica<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Transitiva\" >Transitiva<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Outras-classificacoes\" >Outras classifica\u00e7\u00f5es<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Relacao-inversa\" >Rela\u00e7\u00e3o inversa<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Relacao-Composta\" >Rela\u00e7\u00e3o Composta<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Na-pratica\" >Na pr\u00e1tica!<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-qual-a-definicao-de-relacao-binaria\"><span class=\"ez-toc-section\" id=\"Qual-a-definicao-de-relacao-binaria\"><\/span>Qual a defini\u00e7\u00e3o de rela\u00e7\u00e3o bin\u00e1ria?&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>rela\u00e7\u00e3o entre dois elementos de conjuntos diferentes que seguem a mesma propriedade<\/strong> \u00e9 chamada de Rela\u00e7\u00e3o Bin\u00e1ria. Uma rela\u00e7\u00e3o bin\u00e1ria \ud835\udc45 de \ud835\udc34 em \ud835\udc35 \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/3_I0uF_0eC5G6xJXm92J8_uuc2l7Lb8fC52K1d-sLE4SNYYfXzfHX_oyD0BmiSePQ3lHiG60Oo3q3VoZ3TFNF0djGVF1_NGGBx9PAn1ps8zFb8jGd-YZHjuTNjUOgk6OA5A6tmcuqHwYGEwZIAeBzTZU3WUc0eLepDGOoiFAhmoGzRa1WoXN4GdZUw\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Observe que <strong><em>x<\/em><\/strong><strong> pertence ao conjunto A e <\/strong><strong><em>y<\/em><\/strong><strong>, ao conjunto B.<\/strong> Al\u00e9m disso, \ud835\udc5d(\ud835\udc65, \ud835\udc66) representa o crit\u00e9rio de relacionamento entre os elementos <em>x <\/em>e <em>y, <\/em>ou seja, sua propriedade.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assim, vemos que <strong><em>R <\/em><\/strong><strong>\u00e9 um subconjunto de A x B<\/strong> e podemos afirmar que:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>&nbsp;\ud835\udc45 \u2282 \ud835\udc34 \ud835\udc65 \ud835\udc35;&nbsp;<\/li><li>\ud835\udc34 \u00e9 o conjunto de partida da rela\u00e7\u00e3o \ud835\udc45; e<\/li><li>\ud835\udc35 \u00e9 o conjunto de chegada, ou contradom\u00ednio, da rela\u00e7\u00e3o \ud835\udc45.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Se (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc35 e \ud835\udc5d(\ud835\udc4e, \ud835\udc4f) \u00e9 verdadeira, podemos usar a nota\u00e7\u00e3o:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc4e \ud835\udc45 \ud835\udc4f ou \ud835\udc45(\ud835\udc4e) = \ud835\udc4f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Agora, se queremos dizer que (\ud835\udc50, \ud835\udc51) \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc35 e \ud835\udc5d(\ud835\udc50, \ud835\udc51) \u00e9 falsa:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc50 \ud835\udc45 \ud835\udc51 ou \ud835\udc45(\ud835\udc50) \u2260 \ud835\udc51<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Chamamos de conjunto solu\u00e7\u00e3o de uma rela\u00e7\u00e3o bin\u00e1ria o seguinte:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc46 = {(\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc35|\ud835\udc5d(\ud835\udc4e, \ud835\udc4f) \u00e9 \ud835\udc49}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">O <strong>conjunto \ud835\udc46 possui todos os elementos que satisfazem a propriedade da rela\u00e7\u00e3o<\/strong>.\u00a0<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div id=\"new-form\">\n\t<form id=\"forms_layout\">\n\t<input type=\"hidden\" id=\"chave_de\" name=\"chave_de\" value=\"01_EMIL_LEADS_TOTAIS\" \/>\n\t\t<input type=\"hidden\" id=\"mid\" name=\"mid\" value=\"515009017\" \/>\n\t\t<input type=\"hidden\" id=\"Objetivo\" name=\"Objetivo\" value=\"Militares\" \/>\n\t\t<input type=\"hidden\" id=\"Estado_de_Origem_do_IP\" name=\"Estado_de_Origem_do_IP\"  value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Cidade_de_Origem_do_IP\" name=\"Cidade_de_Origem_do_IP\" value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Modo_de_entrada\" name=\"Modo_de_entrada\" value=\"Portal\" \/>\n\t\t<input type=\"hidden\" id=\"Pagina_Origem\" name=\"Pagina_Origem\" value=\"\" \/>\n\n\t\t<div class=\"texto\">\n\t\t\t<h3 class=\"news\"><span class=\"ez-toc-section\" id=\"Inscreva-se-em-nossa-newsletter\"><\/span>Inscreva-se em nossa newsletter!<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\t\t\t<p>Receba not\u00edcias sobre os mais importantes concursos para as For\u00e7as Armadas brasileiras e informa\u00e7\u00f5es sobre o mundo militar!<\/p>\n\t\t<\/div>\n\t\t<div class=\"campos\">\n\t\t\t<div class=\"first_name labels\">\n\t\t\t\t<input type=\"text\" class=\"input_labels\" id=\"first_name\" name=\"Nome_Completo\" placeholder=\"Nome\" required \/>\n\t\t\t<\/div>\n\t\t\t<div class=\"email labels\">\n\t\t\t\t<input type=\"email\" class=\"input_labels\" id=\"email\" name=\"Email\" placeholder=\"E-mail\" required \/>\n\t\t\t<\/div>\n\t\t\t<button type=\"submit\" class=\"button_layout\">\n\t\t\t\t<div id=\"lblBotao\">Inscreva-se<\/div>\n\t\t\t\t<span id=\"divSpiner\" class=\"loader\" role=\"status\" style=\"display: none;\"><\/span>\t\n\t\t\t<\/button>\n\t\t<\/div><!-- \/campos -->\n\t\t\n\t\t<label class=\"container_checkbox\">Voc\u00ea concorda com a nossa\n\t\t\t<a class=\"politics_privacy_ahref\" target=\"_blank\" href=\"https:\/\/gratis.estrategiaconcursos.com.br\/politica-de-privacidade\/\">Pol\u00edtica de Privacidade<\/a> e aceita receber informa\u00e7\u00f5es adicionais do Estrat\u00e9gia Educacional.\n\t\t\t<input id=\"checkbox_layout\" type=\"checkbox\" checked=\"checked\" required \/>\n\t\t\t<span class=\"checkbox\"><\/span>\n\t\t<\/label>\n\t\t<div id=\"msg\"><\/div>\n\t\t<div id=\"error\" class=\"error\">\n\t\t<\/div>\n\t<\/form>\n<\/div><!-- \/new-form -->\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-dominio-e-imagem\"><span class=\"ez-toc-section\" id=\"Dominio-e-Imagem\"><\/span>Dom\u00ednio e Imagem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Para definirmos o que \u00e9 o dom\u00ednio e a imagem de uma rela\u00e7\u00e3o bin\u00e1ria R, considere que R \u00e9 uma rela\u00e7\u00e3o de A em B com \ud835\udc65 \u2208 \ud835\udc34 e \ud835\udc66 \u2208 \ud835\udc35.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Chamamos de Dom\u00ednio de <\/strong><strong><em>R<\/em><\/strong><strong> o conjunto formado por todos os primeiros elementos dos pares ordenados da rela\u00e7\u00e3o bin\u00e1ria.<\/strong> Tamb\u00e9m \u00e9 poss\u00edvel dizer que s\u00e3o todos valores poss\u00edveis no <strong>eixo das abscissas (x) de um plano cartesiano<\/strong>. Sua nota\u00e7\u00e3o \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Yi3pJU5Ylp8F-TAF8tvlrU21d5HMqpDzTBwAhIlr7RkfQSdl9whM4wYQxUM93JkWJ84HnBjjqbp7fJN2rZapyzsBivGJTyZdu3cUrf9_iQlZ-wDIcRJmgd0ADMwu5PS07-r1iAQDVLwT-XZw9fC4KHc8NNJKSHuAEXoX11k85DQSqUrwqBhncrbUOA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">J\u00e1 a <strong>Imagem de <\/strong><strong><em>R<\/em><\/strong><strong> \u00e9 o conjunto de todos os segundos elementos dos pares ordenados<\/strong> da rela\u00e7\u00e3o, cujos valores s\u00e3o representados no <strong>eixo das ordenadas (y) <\/strong>de um plano cartesiano. Veja a nota\u00e7\u00e3o:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/jDJS7uQpggiDxQ5bCZX3lzirLslc9m5f876VKyH4srexZK98jU_XksshUfYpatmAmLOwwvXJVQcF6BsR5N73fzp9jpz4caFowOVjb8Ol5NVaoWBsDnTSSKserjk7Vu7G0MKxW-8f_xcBMZfnOfIEFxQf_TkQKBRM14BOwP4v1dg9U_ah-Vj6r_QKrQ\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Confira a representa\u00e7\u00e3o gen\u00e9rica de uma rela\u00e7\u00e3o <em>R de<\/em> \ud835\udc34 em \ud835\udc35<em> <\/em>no plano cartesiano. Seja \ud835\udc45 \u2282 \ud835\udc34 \ud835\udc65 \ud835\udc35:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/ulcTNCMyvSZ083lb01xS-VX8Jd3Ym-fESjzUj-jnFOD4WtO1DWkPyRZGVc9WNef9tsPRT-MR5J4MEnzown1K2M5gNaAO5Oy98xeKslAnpY0Fl4jfIt4C24ONQdSIQJ4UZkODex_tnsNAeeeppDQ_G9fyMw6K-jL2Yh5pe0ZzP-u4MginkVC82ePkRQ\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ao observar o exemplo acima, podemos notar que \ud835\udc37<sub>\ud835\udc45<\/sub> \u2282 \ud835\udc34 e \ud835\udc3c\ud835\udc5a<sub>\ud835\udc45<\/sub> \u2282 \ud835\udc35.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-tipos-de-relacao-binaria\"><span class=\"ez-toc-section\" id=\"Tipos-de-Relacao-Binaria\"><\/span>Tipos de Rela\u00e7\u00e3o Bin\u00e1ria&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c9 poss\u00edvel classificar as rela\u00e7\u00f5es bin\u00e1rias em alguns tipos, como:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>reflexiva;&nbsp;<\/li><li>sim\u00e9trica;&nbsp;<\/li><li>antissim\u00e9trica; e&nbsp;<\/li><li>transitiva.&nbsp;<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Vamos explicar um pouco mais cada um deles! Para todas, considere \ud835\udc45 \u2282 \ud835\udc34<sup>2<\/sup>.&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-reflexiva\"><span class=\"ez-toc-section\" id=\"Reflexiva\"><\/span>Reflexiva<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A rela\u00e7\u00e3o R \u00e9 considerada reflexiva em A<sup>2<\/sup> quando \u2200\ud835\udc82 \u2208 \ud835\udc68, temos (\ud835\udc82, \ud835\udc82) \u2208 \ud835\udc79. Como exemplo, pense que <em>A<\/em> = [0, 2] e \ud835\udc45<sub>1<\/sub>, \ud835\udc45<sub>2<\/sub> \u2282 \ud835\udc34<sup>2 <\/sup>representados no plano cartesiano:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/1JJCGVil-hr0pNQA49Jrol6SScGrkNpobcCceKs0IxqMizjbrqqihiMluk7IUUHtPcJ6y1Ec-h4-zG7p3nOuKkiadTY3Gk-ZUfa9lR-T5miiBL5k-mc7u1h6-o7Od8_ou_CxpieJi57gZpQQZG6koQchSN3KMTHgGJi3yPwtCReiS5s2zqRkTuE3QA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Observe que \ud835\udc45<sub>1<\/sub> \u00e9 reflexiva e \ud835\udc45<sub>2<\/sub>, n\u00e3o.<strong> Para que a rela\u00e7\u00e3o bin\u00e1ria seja reflexiva, os elementos do conjunto devem conter a reta \ud835\udc66 = \ud835\udc65<\/strong>. No exemplo, apenas \ud835\udc45<sub>1<\/sub> satisfaz essa condi\u00e7\u00e3o.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-simetrica\"><span class=\"ez-toc-section\" id=\"Simetrica\"><\/span>Sim\u00e9trica<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A rela\u00e7\u00e3o \ud835\udc79 \u00e9 sim\u00e9trica em \ud835\udc68<sup>\ud835\udfd0<\/sup> quando para (\ud835\udc82, \ud835\udc83) \u2208 \ud835\udc79, temos (\ud835\udc83, \ud835\udc82) \u2208 \ud835\udc79. Veja o exemplo abaixo, em que \ud835\udc34 = [0, 2] e \ud835\udc45 \u2282 \ud835\udc34<sup>2<\/sup>:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/blB-HQo8Ed-JAoFs1lW2RExsi6IWlM3mbqzdbdftf3HAniwA-PaB0g_dEKCzffgu8IYrTlkYB_MFwIGYW0hCEiUC52OxZdmOcV8PP4O30apbb0xebdHHL5UcUlwrcGE-J_1ZP7khZJfx8tb6T0dqHEXp29pfH3nWeFRNpfdGDa5LVOEQbbK-WIJyrg\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-antissimetrica\"><span class=\"ez-toc-section\" id=\"Antissimetrica\"><\/span>Antissim\u00e9trica<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A rela\u00e7\u00e3o \ud835\udc79 \u00e9 antissim\u00e9trica em \ud835\udc68<sup>\ud835\udfd0<\/sup> quando (\ud835\udc82, \ud835\udc83) \u2208 \ud835\udc79 e (\ud835\udc83, \ud835\udc82) \u2208 \ud835\udc79, temos \ud835\udc82 = \ud835\udc83. Veja uma representa\u00e7\u00e3o gr\u00e1fica, em que \ud835\udc34 = [0, 2] e \ud835\udc45 \u2282 \ud835\udc34<sup>2<\/sup>:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/6HpCTh2Cr19Natf8Jt1EsQBT8japiHpttc8fxyzdvlivNFnp5TMowfk_ateiNjqcP_Rq_imzjWbbtUnaBLiPtdHXKmUhILCEUXjeuaWloaj5ogaaDy-81NmFTDAeSqvJEYAUeizzcxZxJV1-oUugCtVbhnTEwOucaJpvK4RdlmOBF8hB9_nhuC3k0A\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-transitiva\"><span class=\"ez-toc-section\" id=\"Transitiva\"><\/span>Transitiva<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Agora, no caso de uma rela\u00e7\u00e3o bin\u00e1ria <em>R <\/em>em \ud835\udc68<sup>\ud835\udfd0<\/sup> quando (\ud835\udc82, \ud835\udc83) \u2208 \ud835\udc79 e (\ud835\udc83, \ud835\udc84) \u2208 \ud835\udc79, temos (\ud835\udc82, \ud835\udc84) \u2208 \ud835\udc79. Veja o exemplo para entender melhor:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc45 = {(\ud835\udc65, \ud835\udc66) \u2208 \u2124<sup>2<\/sup>|\ud835\udc65 \u2212 \ud835\udc66 \u22ee 2}<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Para (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45, temos \ud835\udc4e \u2212 \ud835\udc4f = 2\ud835\udc58<sub>1<\/sub>, \ud835\udc58<sub>1<\/sub> \u2208 \u2124.<\/li><li>Para (\ud835\udc4f, \ud835\udc50) \u2208 \ud835\udc45, temos \ud835\udc4f \u2212 \ud835\udc50 = 2\ud835\udc58<sub>2<\/sub>, \ud835\udc58<sub>2<\/sub> \u2208 \u2124.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Somando as duas equa\u00e7\u00f5es, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc4e \u2212 \ud835\udc50 = 2(\ud835\udc58<sub>1<\/sub> + \ud835\udc58<sub>2<\/sub>) = 2\ud835\udc58<sub>3<\/sub><br>\u21d2 (\ud835\udc4e, \ud835\udc50) \u2208 \ud835\udc45<br>\u2234 \ud835\udc45 \u00e9 transitiva<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nesse exemplo, tamb\u00e9m podemos ter outras classifica\u00e7\u00f5es para \ud835\udc45. Veja:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\u2200\ud835\udc4e \u2208 \u2124, \ud835\udc4e \u2212 \ud835\udc4e = 0 = 2 \u2219 0<br>\u21d2 (\ud835\udc4e, \ud835\udc4e) \u2208 \ud835\udc45<br>\u2234 \ud835\udc45 \u00e9 reflexiva<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45, temos \ud835\udc4e \u2212 \ud835\udc4f = 2\ud835\udc58, \ud835\udc58 \u2208 \u2124. Se multiplicamos a equa\u00e7\u00e3o por \u22121, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc4f \u2212 \ud835\udc4e = 2(\u2212\ud835\udc58)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Disso, conclu\u00edmos que (\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45. Ou seja, <strong><em>R<\/em><\/strong><strong> tamb\u00e9m \u00e9 sim\u00e9trica.&nbsp;<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-outras-classificacoes\"><span class=\"ez-toc-section\" id=\"Outras-classificacoes\"><\/span>Outras classifica\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c9 poss\u00edvel que uma rela\u00e7\u00e3o tenha mais de uma classifica\u00e7\u00e3o ao mesmo tempo. Isso resulta em outros dois tipos de rela\u00e7\u00e3o bin\u00e1ria:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\ud835\udc79 \u00e9 uma rela\u00e7\u00e3o de equival\u00eancia quando \u00e9, simultaneamente, reflexiva, sim\u00e9trica e transitiva; e<\/li><li>\ud835\udc79 \u00e9 uma rela\u00e7\u00e3o de ordem quanto \u00e9, simultaneamente, reflexiva, antissim\u00e9trica e transitiva.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Existe ainda uma rela\u00e7\u00e3o espec\u00edfica, chamada de rela\u00e7\u00e3o de identidade. Ela \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">{(\ud835\udc65, \ud835\udc66) \u2208 \ud835\udc34<sup>2<\/sup>|\ud835\udc65 = \ud835\udc66}<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-relacao-inversa\"><span class=\"ez-toc-section\" id=\"Relacao-inversa\"><\/span>Rela\u00e7\u00e3o inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Se quisermos <strong>descobrir o inverso de uma rela\u00e7\u00e3o bin\u00e1ria, precisamos calcular \ud835\udc45<\/strong><strong><sup>\u22121<\/sup><\/strong>, sendo \ud835\udc45 \u2282 \ud835\udc34 \ud835\udc65 \ud835\udc35. Assim, a nota\u00e7\u00e3o se apresenta da seguinte forma:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/160tziMXnQTwMvjU4sz8nmCLfXRS1hffk7wh_1KsAUVQbef8nuwqXkYkdGSxLRbw1Bazi-czlDAXDscQUf65IYMpphW-j6gSEAU_pemoit75MaHtQ1qrWPnfG5JKG--oxvhGK3T_g8ykMp1TM6vh0AAl6kTHXySnvZpA8a1154K0YTBE73LWha49Ww\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>A rela\u00e7\u00e3o inversa \ud835\udc45<\/strong><strong><sup>\u22121<\/sup><\/strong><strong> \u00e9 obtida invertendo-se a ordem dos pares ordenados da rela\u00e7\u00e3o \ud835\udc45.<\/strong> Como exemplo, considere \ud835\udc34 = {\ud835\udc65 \u2208 \u211d|1 \u2264 \ud835\udc65 \u2264 2} e \ud835\udc35 = {\ud835\udc66 \u2208 \u211d|2 \u2264 \ud835\udc66 \u2264 4}. Vamos representar no plano cartesiano as rela\u00e7\u00f5es \ud835\udc45 = {(\ud835\udc65, \ud835\udc66) \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc35|\ud835\udc66 = 2\ud835\udc65} e sua inversa \ud835\udc45<sup>\u22121<\/sup>:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Q5ZzRqxoAInrk4r-DVM35eZhOus0Dkzc-xS4l2rUv4A-7YGFSdRM9UhEDmbKTMEZikbqyucbtAYrY-4gVEJEUDn5YOiaTzkihgLSq2owHBSXiXN3aBCk0-9XAfpRqoEl_K3OXljL1XMAv_EJqMsprJqtbRHTGZGMub6CJp3u_y1VlgRaEMPb9F7Fag\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Perceba que <strong>os gr\u00e1ficos de \ud835\udc45 e \ud835\udc45<\/strong><strong><sup>\u22121<\/sup><\/strong><strong> s\u00e3o sim\u00e9tricos em rela\u00e7\u00e3o \u00e0 reta \ud835\udc66 = \ud835\udc65<\/strong>. Se usarmos a defini\u00e7\u00e3o de \ud835\udc45 ser sim\u00e9trica, temos:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\ud835\udc45 \u00e9 sim\u00e9trica se (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45, temos (\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Por\u00e9m, se vemos a defini\u00e7\u00e3o da inversa de \ud835\udc45:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">(\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45 \u21d2 (\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45<sup>\u22121<\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assim, como \ud835\udc45 \u00e9 sim\u00e9trica, (\ud835\udc4f, \ud835\udc4e) tamb\u00e9m pertence a \ud835\udc45:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">(\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45 \u21d2 (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45<sup>\u22121<\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Logo, <strong>todos os elementos de \ud835\udc45 tamb\u00e9m pertencem a \ud835\udc45<\/strong><strong><sup>\u22121<\/sup><\/strong><strong>.<\/strong> Disso, conclu\u00edmos:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc79 \u00e9 sim\u00e9trica \u21d4 \ud835\udc79 = \ud835\udc79<sup>\u2212\ud835\udfcf<\/sup><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-relacao-composta\"><span class=\"ez-toc-section\" id=\"Relacao-Composta\"><\/span>Rela\u00e7\u00e3o Composta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Outra rela\u00e7\u00e3o que podemos estabelecer \u00e9 a rela\u00e7\u00e3o composta. Ela acontece <strong>quando temos mais do que dois conjuntos<\/strong>, em que \ud835\udc45 \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc35 e \ud835\udc47 \u2208 \ud835\udc35 \ud835\udc65 \ud835\udc36. Ou seja, A \u00e9 o dom\u00ednio e B seu contradom\u00ednio, ao mesmo tempo que B tamb\u00e9m \u00e9 o dom\u00ednio do contradom\u00ednio C.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se \ud835\udc4e \u2208 \ud835\udc34, \ud835\udc4f \u2208 \ud835\udc35, \ud835\udc50 \u2208 \ud835\udc36, temos:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc4e \ud835\udc45 \ud835\udc4f \u21d2 \ud835\udc45(\ud835\udc4e) = \ud835\udc4f<br>\ud835\udc4f \ud835\udc47 \ud835\udc50 \u21d2 \ud835\udc47(\ud835\udc4f) = \ud835\udc50<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Para estabelecer uma rela\u00e7\u00e3o composta entre R e T, temos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc45(\ud835\udc4e) = \ud835\udc4f \ud835\udc52 \ud835\udc47(\ud835\udc4f) = \ud835\udc50 \u21d2 \ud835\udc47(\ud835\udc45(\ud835\udc4e)) = \ud835\udc50<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Obs: <\/strong>\ud835\udc47(\ud835\udc45(\ud835\udc4e)) pode ser reescrita como \ud835\udc47\ud835\udc5c\ud835\udc45(\ud835\udc4e), essa \u00e9 a nota\u00e7\u00e3o usual para representar uma rela\u00e7\u00e3o composta de \ud835\udc45 com \ud835\udc47.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A partir disso, veja como fica a defini\u00e7\u00e3o da rela\u00e7\u00e3o composta:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc47\ud835\udc5c\ud835\udc45 = {(\ud835\udc4e, \ud835\udc50) \u2208 \ud835\udc34 \ud835\udc65 \ud835\udc36|\u2203\ud835\udc4f \u2208 \ud835\udc35 \ud835\udc61\ud835\udc4e\ud835\udc59 \ud835\udc5e\ud835\udc62\ud835\udc52 (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45 \ud835\udc52 (\ud835\udc4f, \ud835\udc50) \u2208 \ud835\udc47}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Confuso? Tamb\u00e9m podemos demonstrar essa rela\u00e7\u00e3o de forma mais visual por meio do diagrama de flechas.&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/n72_SUjBZHKqoCN2xQ3SjhLgLUCv4YWAGJbx_PQ9367lbeO8_-RdQoKOeI37ebpEmJWloJ2tz3oD1Q8kHAWt0SF1rNb6i81ZiOAfo7b0x4JWajsnAn3sinhOcFjtzoRPGk8_4u30S26PrjF4IXsuGrk8yTbwrqAAFfyjG1ayoORMiJdRbhJHHX6qTA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Aten\u00e7\u00e3o! <\/strong>Um erro muito comum ao ver a nomenclatura \ud835\udc47\ud835\udc5c\ud835\udc45 de \ud835\udc34 em \ud835\udc36 \u00e9 inverter a ordem das rela\u00e7\u00f5es devido a \ud835\udc47 estar escrito antes de \ud835\udc45.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Para solucionar esse problema, devemos entender que \ud835\udc47\ud835\udc5c\ud835\udc45 = \ud835\udc47(\ud835\udc45(\ud835\udc65)). Assim, vemos que<strong> \ud835\udc65 nos levar\u00e1 a \ud835\udc45(\ud835\udc65) e com \ud835\udc45(\ud835\udc65) encontraremos \ud835\udc47(\ud835\udc45(\ud835\udc65))<\/strong>. Ou, se a quest\u00e3o envolver um diagrama de flechas, basta lembrar que na rela\u00e7\u00e3o composta as rela\u00e7\u00f5es iniciam da direita para a esquerda:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/6AEb6hnG5SMdvSP5myjBVYpVeEbO3nh-LehC3gG3PX7unKCgqKLZ5n81oS-9tFjzgRvfybLSNq4TwaBVIRP7wOEXz3JcTRfR4ARi2mcRJrC5L7N_KBuDC1FIIXLv08ehKW1I1Y9rO4q-WXEjlCUszYnjH98OByffAh34ATHNjlWmyuaAz096Fj12Dg\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-na-pratica\"><span class=\"ez-toc-section\" id=\"Na-pratica\"><\/span>Na pr\u00e1tica!<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Que tal ver um exerc\u00edcio comentado de rela\u00e7\u00e3o composta para fixar melhor o conte\u00fado? Sejam os conjuntos <strong>\ud835\udc34 = {1, 2, 3, 4, 5}, \ud835\udc35 = {\ud835\udc4e, \ud835\udc4f, \ud835\udc50, \ud835\udc51} e \ud835\udc36 = {1, 2, \ud835\udc4e, \ud835\udc4f<\/strong>}. Encontre a rela\u00e7\u00e3o composta \ud835\udc47\ud835\udc5c\ud835\udc45, sendo as rela\u00e7\u00f5es \ud835\udc45 e \ud835\udc47 representadas pelo diagrama abaixo:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/QkFh5bFelbEhPlyvTx9jxMuV1AiMsQA_Ip-Z1TKIFHWEatNXCtJYOY_g-WQ724uEqo7muRFHfaaydSbsfl8K4Lmopk6MyeSV2mkNUeK1aRNQaWcX7rBqBuQCNVb9shO-F5e4TbERdvLtioF2sgG1UWxXcq2Q83x_rgdVEw2YXtpbxLBk9RcxDKYB8g\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ao observar o diagrama, lembramos que <strong>os elementos que possuem flechas saindo deles ser\u00e3o os primeiros elementos de um par ordenado e os elementos indicados por estas flechas ser\u00e3o os segundos elementos<\/strong> desse par ordenado.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">No caso da rela\u00e7\u00e3o <em>R<\/em>, vemos que o elemento 1 possui uma flecha saindo dela e esta chega em \ud835\udc4e e \ud835\udc4f. Dessa forma, podemos escrever:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">(1, \ud835\udc4e), (1, \ud835\udc4f) \u2208 \ud835\udc45<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Analogamente para os outros elementos:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">(3, \ud835\udc50), (5, \ud835\udc51) \u2208 \ud835\udc45<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assim, o conjunto <em>R<\/em> \u00e9 dado por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc45 = {(1, \ud835\udc4e), (1, \ud835\udc4f), (3, \ud835\udc50), (5, \ud835\udc51)}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Aplicando a mesma l\u00f3gica para a rela\u00e7\u00e3o <em>T<\/em>, temos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc47 = {(\ud835\udc4e, 1), (\ud835\udc50, 2), (\ud835\udc50, \ud835\udc4e), (\ud835\udc51, \ud835\udc4e), (\ud835\udc51, \ud835\udc4f)}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Agora, se quisermos estabelecer uma rela\u00e7\u00e3o composta, devemos perceber que<strong> \ud835\udc47\ud835\udc5c\ud835\udc45 leva elementos do conjunto \ud835\udc34 a elementos do conjunto \ud835\udc36<\/strong>. Para encontrar quais s\u00e3o os pares ordenados dessa rela\u00e7\u00e3o, devemos analisar o caminho que as flechas percorrem.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">O elemento 1 do conjunto \ud835\udc34 chega no elemento \ud835\udc4e do conjunto \ud835\udc35. Esse elemento \ud835\udc4e chega no elemento 1 do conjunto \ud835\udc36. Ent\u00e3o, o elemento 1 do conjunto \ud835\udc34 chega no elemento 1 do conjunto \ud835\udc36. Assim, (1, 1) \u2208 \ud835\udc47\ud835\udc5c\ud835\udc45.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Por outro lado, o elemento 1 do conjunto \ud835\udc34 tamb\u00e9m chega no elemento \ud835\udc4f do conjunto \ud835\udc35. Por\u00e9m, esse \ud835\udc4f n\u00e3o chega em nenhum elemento do conjunto \ud835\udc36. Ent\u00e3o, <strong>esse caminho n\u00e3o retorna nenhum elemento da rela\u00e7\u00e3o composta.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Aplicando a mesma l\u00f3gica para os demais elementos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">3 \u2192 \ud835\udc50 \u2192 2 \u21d2 (3, 2) \u2208 \ud835\udc47\ud835\udc5c\ud835\udc45<br>3 \u2192 \ud835\udc50 \u2192 \ud835\udc4e \u21d2 (3, \ud835\udc4e) \u2208 \ud835\udc47\ud835\udc5c\ud835\udc45<br>5 \u2192 \ud835\udc51 \u2192 \ud835\udc4e \u21d2 (5, \ud835\udc4e) \u2208 \ud835\udc47\ud835\udc5c\ud835\udc45<br>5 \u2192 \ud835\udc51 \u2192 \ud835\udc4f \u21d2 (5, \ud835\udc4f) \u2208 \ud835\udc47\ud835\udc5c\ud835\udc45<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assim, a rela\u00e7\u00e3o composta \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\ud835\udc47\ud835\udc5c\ud835\udc45 = {(1, 1), (3, 2), (3, \ud835\udc4e), (5, \ud835\udc4e), (5, \ud835\udc4f)}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E a\u00ed, guerreiro? Curtiu mais este resumo de Matem\u00e1tica? Encontre mais conte\u00fados no Portal <span id=\"urn:enhancement-89ca1243-ae4e-4538-b8ea-5b99dd5513b4\" class=\"textannotation disambiguated wl-organization wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110246\/entity\/estrategia_militares\">Estrat\u00e9gia Militares<\/span>, al\u00e9m de not\u00edcias sobre os seletivos das <span id=\"urn:enhancement-2731351e-e978-43fe-a360-9ec826106aae\" class=\"textannotation disambiguated wl-creative-work wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110246\/entity\/forcas_armadas\">For\u00e7as Armadas<\/span>! Se voc\u00ea quiser treinar os conhecimentos que aprendeu, conhe\u00e7a tamb\u00e9m o nosso Banco de Quest\u00f5es do <span id=\"urn:enhancement-b5825c54-9ef0-4b65-8e4f-13b9b4dcb2de\" class=\"textannotation disambiguated wl-organization wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110246\/entity\/estrategia_militares\">Estrat\u00e9gia Militares<\/span> &#8211; s\u00e3o milhares de quest\u00f5es in\u00e9ditas e de provas anteriores para voc\u00ea treinar!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\">Progress\u00e3o Aritm\u00e9tica (PA): o que \u00e9, tipos e propriedades<\/a><\/li><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/\">Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais<\/a><\/li><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\">Par ordenado e produto cartesiano: o que s\u00e3o e suas propriedades<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"Se voc\u00ea tem o sonho de ingressar em algum dos seletivos das For\u00e7as Armadas, sabe que a mat\u00e9ria&hellip;\n","protected":false},"author":9,"featured_media":60302,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[307],"tags":[],"class_list":["post-60301","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO 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