{"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>Se voc\u00ea tem o sonho de ingressar em algum dos seletivos das For\u00e7as Armadas, 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_79_2 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>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>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>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>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\">\ud835\udc4e \ud835\udc45 \ud835\udc4f ou \ud835\udc45(\ud835\udc4e) = \ud835\udc4f<\/p>\n\n\n\n<p>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\">\ud835\udc50 \ud835\udc45 \ud835\udc51 ou \ud835\udc45(\ud835\udc50) \u2260 \ud835\udc51<\/p>\n\n\n\n<p>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\">\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>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><\/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>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><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>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>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>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>\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>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>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>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>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>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>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\">\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>Somando as duas equa\u00e7\u00f5es, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\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>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\">\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>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\">\ud835\udc4f \u2212 \ud835\udc4e = 2(\u2212\ud835\udc58)<\/p>\n\n\n\n<p>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>\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>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\">{(\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>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><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>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>Por\u00e9m, se vemos a defini\u00e7\u00e3o da inversa de \ud835\udc45:<\/p>\n\n\n\n<p class=\"has-text-align-center\">(\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45 \u21d2 (\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45<sup>\u22121<\/sup><\/p>\n\n\n\n<p>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\">(\ud835\udc4f, \ud835\udc4e) \u2208 \ud835\udc45 \u21d2 (\ud835\udc4e, \ud835\udc4f) \u2208 \ud835\udc45<sup>\u22121<\/sup><\/p>\n\n\n\n<p>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\">\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>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>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\">\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>Para estabelecer uma rela\u00e7\u00e3o composta entre R e T, temos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\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><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>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\">\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>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><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>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>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>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>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\">(1, \ud835\udc4e), (1, \ud835\udc4f) \u2208 \ud835\udc45<\/p>\n\n\n\n<p>Analogamente para os outros elementos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">(3, \ud835\udc50), (5, \ud835\udc51) \u2208 \ud835\udc45<\/p>\n\n\n\n<p>Assim, o conjunto <em>R<\/em> \u00e9 dado por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc45 = {(1, \ud835\udc4e), (1, \ud835\udc4f), (3, \ud835\udc50), (5, \ud835\udc51)}<\/p>\n\n\n\n<p>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\">\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>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>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>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>Aplicando a mesma l\u00f3gica para os demais elementos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">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>Assim, a rela\u00e7\u00e3o composta \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\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>E a\u00ed, guerreiro? Curtiu mais este resumo de Matem\u00e1tica? Encontre mais conte\u00fados no Portal Estrat\u00e9gia Militares, al\u00e9m de not\u00edcias sobre os seletivos das For\u00e7as Armadas! Se voc\u00ea quiser treinar os conhecimentos que aprendeu, conhe\u00e7a tamb\u00e9m o nosso Banco de Quest\u00f5es do Estrat\u00e9gia Militares &#8211; s\u00e3o milhares de quest\u00f5es in\u00e9ditas e de provas anteriores para voc\u00ea treinar!<\/p>\n\n\n\n<p><\/p>\n\n\n\n\n\n<p><\/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":{"wl_entities_gutenberg":"","footnotes":""},"categories":[307],"tags":[],"wl_entity_type":[187],"class_list":{"0":"post-60301","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>O que \u00e9 uma 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