{"id":32327,"date":"2021-05-09T13:00:17","date_gmt":"2021-05-09T13:00:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32327"},"modified":"2025-03-03T04:13:53","modified_gmt":"2025-03-03T04:13:53","slug":"dominios-de-integridade-e-os-numeros-inteiros","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/dominios-de-integridade-e-os-numeros-inteiros\/","title":{"rendered":"Dom\u00ednios de Integridade e os N\u00fameros Inteiros"},"content":{"rendered":"<style>\n\tp, ul, ol{\n\ttext-align: justify;\n\t}\n\th1{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\t}\n\th2{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\tfont-size:24pt;\n\t}\n\th3 { \n\t\ttext-align: center;\n\t\ttext-transform: uppercase;\n\t\tfont-size: 24px !important;\n\t}\n<\/style>\n<h1>Dom\u00ednios de Integridade e os N\u00fameros Inteiros<\/h1>\n<p style=\"text-align:center;\"><em><strong>Resumo:<\/strong><br \/>\n\tNesta aula, \u00e9 introduzido o conceito de Dom\u00ednio de Integridade, explicada sua relev\u00e2ncia no estudo da \u00e1lgebra geral e demonstradas, por meio de provas formais, algumas de suas propriedades mais importantes. <\/em><\/p>\n<p style=\"text-align:center;\"><em><strong>Objetivos de Aprendizagem:<\/strong><\/em><br \/>\nAo concluir esta aula, o estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> o prop\u00f3sito do estudo da \u00e1lgebra geral. <\/li>\n<li><strong>Compreender<\/strong> o conceito de dom\u00ednio de integridade. <\/li>\n<li><strong>Explicar<\/strong> os aspectos b\u00e1sicos comuns entre os dom\u00ednios de integridade e os n\u00fameros inteiros. <\/li>\n<li><strong>Demonstrar<\/strong> por meio de provas formais as propriedades b\u00e1sicas dos dom\u00ednios de integridade. <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n\t<strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u><\/strong><br \/>\n\t<a href=\"#1\">O OBJETIVO DA \u00c1LGEBRA GERAL E CONHECIMENTOS PR\u00c9VIOS<\/a><br \/>\n\t<a href=\"#2\">DOS N\u00daMEROS INTEIROS AOS DOM\u00cdNIOS DE INTEGRIDADE<\/a><br \/>\n\t<a href=\"#3\">ASPECTOS B\u00c1SICOS COMUNS AOS DOM\u00cdNIOS DE INTEGRIDADE E AOS N\u00daMEROS INTEIROS<\/a><br \/>\n\t<a href=\"#4\">PROPRIEDADES DOS DOM\u00cdNIOS DE INTEGRIDADE E DOS N\u00daMEROS INTEIROS<\/a><br \/>\n\t<a href=\"#5\">EXERC\u00cdCIOS<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hxmc1-eXWxU?si=57GADT52JG4fHFT-\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>O objetivo da \u00e1lgebra geral e conhecimentos pr\u00e9vios<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=183s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">O objetivo principal da \u00e1lgebra geral<\/span><\/strong><\/a> \u00e9 o estudo de toda a variedade de sistemas matem\u00e1ticos poss\u00edveis. Aqui estudaremos v\u00e1rios desses sistemas, e entre os mais importantes destacam-se os n\u00fameros naturais e inteiros, e atrav\u00e9s destes \u00faltimos chegaremos aos dom\u00ednios de integridade.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{N}= \\{1,2,3,4,\\cdots\\}<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}= \\{0,\\pm 1,\\pm 2,\\pm 3,\\pm 4,\\cdots\\}<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Dos n\u00fameros inteiros aos dom\u00ednios de integridade<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=358s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Come\u00e7aremos nosso estudo com os n\u00fameros inteiros,<\/span><\/strong><\/a> e a raz\u00e3o para proceder dessa maneira \u00e9 que eles possuem a maior quantidade de semelhan\u00e7as com a maioria dos sistemas num\u00e9ricos que analisaremos neste estudo.<\/p>\n<p>Em vez de tentar definir o que s\u00e3o os n\u00fameros inteiros, iniciaremos supondo que, seja o que for, eles satisfazem certas propriedades. Para isso, escolhe-se um conjunto de axiomas de modo que seja poss\u00edvel inferir todas as propriedades que intuitivamente associamos aos inteiros.<\/p>\n<p>Todas essas defini\u00e7\u00f5es s\u00e3o feitas por meio dos <strong>axiomas de Peano<\/strong> dos Naturais, ao introduzir as opera\u00e7\u00f5es b\u00e1sicas da aritm\u00e9tica. Seguindo esse m\u00e9todo axiom\u00e1tico e ampliando as diferentes opera\u00e7\u00f5es sobre os naturais e inteiros, obtemos novos conjuntos num\u00e9ricos, como os racionais, irracionais, reais, complexos, quaterni\u00f5es, octoni\u00f5es, e muitos outros.<\/p>\n<p>Depois, ao observarmos os n\u00fameros inteiros, veremos que eles possuem propriedades que se repetem em todos os demais conjuntos num\u00e9ricos, como a exist\u00eancia de um elemento neutro multiplicativo, um elemento neutro aditivo e leis distributivas. Assim, ao nos referirmos a essas propriedades, podemos estabelecer uma linguagem que nos permita falar sobre todos esses conjuntos simultaneamente. \u00c9 nesse contexto que surgem termos como:<\/p>\n<ul>\n<li>Dom\u00ednio de Integridade<\/li>\n<li>Anel<\/li>\n<li>Grupo<\/li>\n<li>Espa\u00e7o Vetorial<\/li>\n<\/ul>\n<p>E uma longa lista de outros termos desse tipo. N\u00f3s concentraremos nossos esfor\u00e7os no estudo dos <strong>Dom\u00ednios de Integridade.<\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Aspectos b\u00e1sicos comuns aos dom\u00ednios de integridade e aos n\u00fameros inteiros<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=472s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Para explicar o que \u00e9 um dom\u00ednio de integridade<\/span><\/strong><\/a> utilizaremos as propriedades que compreendemos muito bem a partir dos n\u00fameros inteiros. Nesse contexto, temos que, se <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> s\u00e3o n\u00fameros inteiros, ent\u00e3o s\u00e3o v\u00e1lidas as <strong>leis<\/strong><\/p>\n<ol>\n<li><strong>Comutativas:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = b + a<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab = ba<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Associativas:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a+b+c = (a+b)+c<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(ab)c = abc = a(bc)<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Distributivas:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a(b+c) = ab+ac<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>Al\u00e9m disso, existem certos elementos especiais conhecidos como neutros:<\/p>\n<ul>\n<li><strong>Neutro aditivo:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = a \\leftrightarrow c=0<\/span><\/span><\/li>\n<li><strong>Neutro multiplicativo:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac = a \\leftrightarrow c=1<\/span><\/span><\/li>\n<\/ul>\n<p>O objeto cujo s\u00edmbolo \u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> \u00e9 o neutro aditivo, e aquele que corresponde ao s\u00edmbolo <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> \u00e9 o neutro multiplicativo.<\/p>\n<p>Os inteiros tamb\u00e9m possuem inversos aditivos. A cada n\u00famero inteiro corresponde um inverso aditivo que, ao ser somado com ele, resulta no neutro aditivo.<\/p>\n<ul>\n<li><strong>Inverso aditivo:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = 0 \\longleftrightarrow c=-a<\/span><\/span><\/li>\n<\/ul>\n<p>Os inversos aditivos s\u00e3o reconhecidos pelo sinal \u00ab-\u00bb que os acompanha.<\/p>\n<p>Finalmente, existe uma <strong>lei de simplifica\u00e7\u00e3o<\/strong> que \u00e9 expressa pela rela\u00e7\u00e3o<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(c\\neq 0 \\wedge ca = cb) \\longleftrightarrow (a=b)<\/span><\/span><\/p>\n<p>Essas propriedades que revisamos s\u00e3o v\u00e1lidas para muitos outros conjuntos: reais, complexos, polin\u00f4mios, etc. Dessa forma, chamamos de <strong>Dom\u00ednio de Integridade<\/strong> todos os conjuntos que satisfazem essas propriedades.<\/p>\n<p><span style=\"color: #800000;\"><strong>DEFINI\u00c7\u00c3O:<\/strong><\/span> Um Dom\u00ednio de Integridade \u00e9 qualquer conjunto <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> provido de uma opera\u00e7\u00e3o de soma e produto tais que<\/p>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow a+b \\in D<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow ab \\in D<\/span><\/span><\/li>\n<\/ul>\n<p>E al\u00e9m disso, satisfazem-se as leis <strong>associativas, comutativas<\/strong> e <strong>distributiva,<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> cont\u00e9m <strong>neutros aditivos<\/strong> e <strong>multiplicativos<\/strong> (cada um deles \u00e9 \u00fanico) e, finalmente, vale a <strong>lei de simplifica\u00e7\u00e3o.<\/strong><\/p>\n<h4>Exemplo de Dom\u00ednio de Integridade<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=749s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consideremos o conjunto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\{a+b\\sqrt{3}\\; |\\; a,b\\in \\mathbb{Z}\\}.<\/span><\/span><\/span> <\/strong><\/a>Este conjunto, munido das opera\u00e7\u00f5es usuais de soma e produto, \u00e9 um dom\u00ednio de integridade porque satisfaz as leis de comutatividade, associatividade e distributividade, possui neutro aditivo e multiplicativo, e, finalmente, um inverso aditivo.<\/p>\n<ul>\n<li><strong>Neutro aditivo:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Neutro multiplicativo:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Inverso aditivo:<\/strong> Todo elemento <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b\\sqrt{3}<\/span><\/span> tem inverso aditivo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-a-b\\sqrt{3}<\/span><\/span><\/li>\n<\/ul>\n<p>E o mais importante de tudo. Este conjunto A \u00e9 fechado para as opera\u00e7\u00f5es de soma e produto, no sentido de que, se tomamos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span>, ent\u00e3o teremos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x+y\\in A<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xy\\in A.<\/span><\/span> Isso \u00e9 f\u00e1cil de verificar: Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_1 + b_1\\sqrt{3}<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_2 + b_2\\sqrt{3}<\/span><\/span> s\u00e3o elementos de <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, ent\u00e3o teremos que<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\t(a_1 + b_1\\sqrt{3}) + (a_2 + b_2\\sqrt{3}) &amp;=(a_1+a_2) + (b_1 + b_2)\\sqrt{3} \\in A\\\\ \\\\\n\n\t(a_1 + b_1\\sqrt{3})  (a_2 + b_2\\sqrt{3})  &amp;= a_1a_2 + a_1b_2\\sqrt{3}+b_1a_2\\sqrt{3} + 3b_1b_2 \\\\\n\n\t&amp;=(a_1a_2 + 3b_1b_2) + (a_1b_2 + b_1a_2)\\sqrt{3} \\in A\n\n\t\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/y6jXNPhjKv4?si=9SaXhWHN42sC73lZ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Propriedades dos Dom\u00ednios de Integridade e dos N\u00fameros Inteiros<\/h3>\n<h4>O neutro aditivo de um dom\u00ednio de integridade \u00e9 \u00fanico<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=26s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Isso pode ser demonstrado por redu\u00e7\u00e3o ao absurdo:<\/span><\/strong><\/a> Suponhamos que existam dois neutros aditivos, sejam <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0^\\prime<\/span><\/span> tais neutros. Ent\u00e3o teremos que:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 0\\neq 0^\\prime &amp; \\text{; Premissa}\\\\\n\n(2) &amp; a+0 = a &amp; \\text{; Premissa: $0$ \u00e9 neutro aditivo}\\\\\n\n(3) &amp; b+0^\\prime = b &amp; \\text{; Premissa: $0^\\prime$ \u00e9 neutro aditivo}\\\\\n\n(4) &amp; 0^\\prime + 0 = 0^\\prime &amp; \\text{; Substituindo $a=0^\\prime$ em $(2)$}\\\\\n\n(5) &amp; 0 + 0^\\prime = 0 &amp; \\text{; Substituindo $b=0$ em $(3)$}\\\\\n\n(6) &amp; 0 = 0^\\prime  &amp; \\text{; De$(4,5)$ e comutatividade da soma}\\\\\n\n(7) &amp; \\bot &amp;\\text{; De$(1,6)$}\n\n\\end{array}<\/span>\n<p>A partir desse racioc\u00ednio, conclu\u00edmos que, portanto:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0 \\neq 0^\\prime, a + 0 = a, b + 0^\\prime = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Logo, por redu\u00e7\u00e3o ao absurdo, temos<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{a + 0 = a, b + 0^\\prime = b\\}\\vdash 0 = 0^\\prime.<\/span><\/span><\/p>\n<p>Ou seja, se h\u00e1 dois neutros aditivos, ent\u00e3o eles s\u00e3o o mesmo, e, portanto, \u00e9 \u00fanico.<\/p>\n<h4>O neutro multiplicativo tamb\u00e9m \u00e9 \u00fanico<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=305s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A demonstra\u00e7\u00e3o \u00e9 praticamente an\u00e1loga \u00e0 anterior.<\/span><\/strong><\/a> Se existissem dois: <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^\\prime<\/span><\/span>, ent\u00e3o poder\u00edamos fazer o seguinte racioc\u00ednio:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 1\\neq 1^\\prime &amp; \\text{; Premissa}\\\\\n\n(2) &amp; 1\\cdot a = a &amp; \\text{; Premissa: $1$ \u00e9 neutro multiplicativo}\\\\\n\n(3) &amp; 1^\\prime \\cdot b = b &amp; \\text{; Premissa: $1^\\prime$ \u00e9 neutro multiplicativo}\\\\\n\n(4) &amp; 1\\cdot 1^\\prime = 1^\\prime &amp; \\text{; Substituindo $a=1^\\prime$ em $(2)$}\\\\\n\n(5) &amp; 1^\\prime \\cdot 1 = 1 &amp; \\text{; Substituindo $b=1$ em $(3)$}\\\\\n\n(6) &amp; 1 = 1^\\prime  &amp; \\text{; De$(4,5)$ e comutatividade da multiplica\u00e7\u00e3o}\\\\\n\n(7) &amp; \\bot &amp;\\text{; De$(1,6)$}\n\n\\end{array}<\/span>\n<p>Assim, conclu\u00edmos que:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{1 \\neq 1^\\prime, 1a= a, 1b = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Logo, por redu\u00e7\u00e3o ao absurdo, temos que<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{1a= a, 1b= b\\}\\vdash 1 = 1^\\prime.<\/span><\/span><\/p>\n<p>Ou seja, se h\u00e1 dois neutros multiplicativos, ent\u00e3o eles s\u00e3o o mesmo, e, portanto, \u00e9 \u00fanico.<\/p>\n<h4>Vale a lei de simplifica\u00e7\u00e3o para as somas<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Isso \u00e9 o que fazemos quando<\/span><\/strong><\/a> eliminamos termos em uma igualdade<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = a+c \\longleftrightarrow a = c<\/span><\/span><\/p>\n<p>N\u00e3o \u00e9 dif\u00edcil demonstrar essa situa\u00e7\u00e3o, basta seguir o seguinte racioc\u00ednio:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a+b = a+c &amp; \\text{; Premissa} \\\\\n\n(2) &amp; a+b-a = a+c-a &amp; \\text{; De$(1)$, somando $-a$ a ambos os lados} \\\\\n\n(3) &amp; (a-a)+b = (a-a)+c &amp; \\text{; De$(2)$, comutatividade e associatividade} \\\\\n\n(4) &amp; 0+b = 0+c &amp; \\text{; De$(3)$ e Inverso Aditivo} \\\\\n\n(5) &amp; b = c &amp; \\text{; De$(4)$ e Neutro Aditivo} \\\\\n\n\\end{array}<\/span>\n<p>Como esse racioc\u00ednio pode ser feito de ida e volta aplicando os mesmos passos, temos que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\dashv \\vdash b=c<\/span><\/span><\/p>\n<p>O que \u00e9 equivalente a dizer que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash a+b=a+c \\longleftrightarrow b=c<\/span><\/span><\/p>\n<h4>O neutro aditivo \u00e9, por sua vez, um absorvente multiplicativo<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=632s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Com isso simplesmente queremos<\/span><\/strong><\/a> dizer que, para todo <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> no dom\u00ednio de integridade, ser\u00e1 v\u00e1lido que<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 0 = 0<\/span><\/span><\/p>\n<p>Isso tamb\u00e9m \u00e9 f\u00e1cil de demonstrar, basta seguir o seguinte racioc\u00ednio:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+0) &amp; \\text{; Leis distributivas}\\\\\n\n(2) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+a-a) &amp; \\text{; De$(1)$ e Inverso Aditivo}\\\\\n\n(3) &amp; a\\cdot a + a\\cdot 0 = a\\cdot a + a\\cdot a - a\\cdot a &amp; \\text{; De$(2)$ e Distributividade}\\\\\n\n(4) &amp;  a\\cdot 0 =  a\\cdot a - a\\cdot a &amp; \\text{; De$(3)$ e Simplifica\u00e7\u00e3o de somas}\\\\\n\n(5) &amp;  a\\cdot 0 =  0 &amp; \\text{; De$(4)$ e Inverso Aditivo}\\\\\n\n\\end{array}<\/span>\n<h4><strong>Lei dos sinais:<\/strong><\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=736s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">O produto de quantidades de mesmo sinal<\/span><\/strong><\/a> \u00e9 sempre positivo; o produto de quantidades com sinais opostos \u00e9 sempre negativo. A demonstra\u00e7\u00e3o dessa propriedade tamb\u00e9m \u00e9 simples:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot b = a\\cdot b + 0 &amp; \\text{; Neutro Aditivo}\\\\\n\n(2) &amp; a\\cdot b = a\\cdot b + (a)\\cdot(-b) - (a)\\cdot(-b) &amp; \\text{; De$(1)$ e Inverso Aditivo}\\\\\n\n(3) &amp; a\\cdot b = a\\cdot (b -b) - (a)\\cdot(-b) &amp; \\text{; De$(2)$ e Inverso Aditivo}\\\\\n\n(4) &amp; a\\cdot b = a\\cdot 0 + (-a)\\cdot(-b) &amp; \\text{; De$(3)$ e Inverso Aditivo}\\\\\n\n(5) &amp; a\\cdot b = (-a)\\cdot(-b) &amp; \\text{; De$(4)$ e Absorvente Multiplicativo}\\\\\n\n\\end{array}<\/span>\n<p>Portanto: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ab = (-a)(-b)<\/span><\/span><\/p>\n<p>Para os sinais opostos, o racioc\u00ednio \u00e9 semelhante:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot(-b) = a \\cdot (-b) + 0 &amp; \\text{; Neutro Aditivo} \\\\\n\n(2) &amp; a\\cdot(-b) = a \\cdot (-b) + a \\cdot b  -  a \\cdot b  &amp; \\text{; De$(1)$ e Inverso Aditivo} \\\\\n\n(3) &amp; a\\cdot(-b) = a \\cdot (b-b)  -  a \\cdot b  &amp; \\text{; De$(2)$ e Distributividade} \\\\\n\n(4) &amp; a\\cdot(-b) = a \\cdot 0  -  a \\cdot b  &amp; \\text{; De$(3)$ e Inverso Aditivo} \\\\\n\n(5) &amp; a\\cdot(-b) = - a \\cdot b  &amp; \\text{; De$(4)$ e Absorvente Multiplicativo} \\\\\n\n\\end{array}<\/span>\n<p>Portanto: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b) = -a(b)<\/span><\/span><\/p>\n<h4>Se o produto de dois n\u00fameros \u00e9 zero, pelo menos um deles \u00e9 zero<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Outra propriedade que tamb\u00e9m<\/span><\/strong><\/a> \u00e9 amplamente utilizada \u00e9 a seguinte:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=0 \\leftrightarrow (a=0 \\vee b=0)<\/span><\/span><\/p>\n<p>Sua demonstra\u00e7\u00e3o tamb\u00e9m \u00e9 simples:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{a=0\\} \\models a\\cdot b = 0  &amp; \\textbf{; Absorvente Multiplicativo} \\\\\n\n(2) &amp; \\models a=0 \\rightarrow a\\cdot b = 0  &amp;\\text{; TD$(1)$} \\\\\n\n(3) &amp; \\models \\neg (a\\cdot b = 0 ) \\rightarrow \\neg(a=0) &amp;\\text{; CPI$(2)$} \\\\\n\n(4) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(a=0) &amp;\\text{; RTD$(3)$} \\\\\n\n(5) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(b=0) &amp;\\text{; An\u00e1logo$(4)$} \\\\\n\n(6) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models  \\neg(a=0) \\wedge \\neg(b=0) &amp;\\text{; $\\wedge$-int$(4,5)$} \\\\\n\n(7) &amp; \\models (\\neg (a\\cdot b = 0 )) \\rightarrow \\neg(a=0) \\wedge \\neg(b=0)  &amp;\\text{; TD(6)} \\\\\n\n(8) &amp; \\models \\neg(\\neg(a=0) \\wedge \\neg(b=0) ) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; CPI(7)} \\\\\n\n(9) &amp; \\models (a=0 \\vee b=0) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; DM(8)} \\\\\n\n(10)&amp; \\{a\\neq 0 , a\\cdot b=0\\} \\models b=0 &amp; \\textbf{; Absorvente Multiplicativo}\\\\\n\n(11)&amp; \\{a\\cdot b=0\\} \\models a\\neq 0 \\rightarrow  b=0 &amp; \\text{; TD(10)}\\\\\n\n(12)&amp; \\{a\\cdot b=0\\} \\models \\neg(a\\neq 0) \\vee  b=0 &amp; \\text{; $\\rightarrow$-Def(11)}\\\\\n\n(13)&amp; \\{a\\cdot b=0\\} \\models a=0 \\vee  b=0 &amp; \\text{; DN(12)}\\\\\n\n(14)&amp; \\models (a\\cdot b=0) \\rightarrow (a=0 \\vee  b=0) &amp; \\text{; TD(13)}\\\\\n\n(15)&amp; \\models (a\\cdot b=0) \\leftrightarrow (a=0 \\vee  b=0) &amp; \\text{; De(9,14)}\n\n\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KZQ3PXeMlKk?si=O_Hek5KFG853Q6qT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exerc\u00edcios<\/h2>\n<p>Sejam <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> elementos quaisquer de um dom\u00ednio de integridade <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>. Demonstre que as seguintes propriedades s\u00e3o v\u00e1lidas:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(-a)=(-1)a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(a+b)=(-a) + (-b)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=827s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b)=-(ab)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1213s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(-a)=a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1628s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(b-c) = ab - ac<\/span><\/span> <strong>[PROPOSTO]<\/strong><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)+(b-c) = a-c<\/span><\/span> <strong>[PROPOSTO]<\/strong><\/li>\n<li>Para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in D<\/span><\/span>, existe um \u00fanico <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 1 = a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=2029s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xx = x \\leftrightarrow (x=1 \\vee x=0)<\/span><\/span> <strong>[PROPOSTO]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Dom\u00ednios de Integridade e os N\u00fameros Inteiros Resumo: Nesta aula, \u00e9 introduzido o conceito de Dom\u00ednio de Integridade, explicada sua relev\u00e2ncia no estudo da \u00e1lgebra geral e demonstradas, por meio de provas formais, algumas de suas propriedades mais importantes. Objetivos de Aprendizagem: Ao concluir esta aula, o estudante ser\u00e1 capaz de: Compreender o prop\u00f3sito do [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32317,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[1033,571],"tags":[],"class_list":["post-32327","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-geral","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Dom\u00ednios de Integridade e os N\u00fameros Inteiros - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explore os Dom\u00ednios de Integridade na \u00e1lgebra, suas propriedades e sua rela\u00e7\u00e3o com os n\u00fameros inteiros atrav\u00e9s de demonstra\u00e7\u00f5es e exerc\u00edcios.\" \/>\n<meta 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