{"id":27490,"date":"2021-02-14T13:00:26","date_gmt":"2021-02-14T13:00:26","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27490"},"modified":"2024-07-18T20:58:58","modified_gmt":"2024-07-18T20:58:58","slug":"leis-de-demorgan-distribuicao-e-suas-provas","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/leis-de-demorgan-distribuicao-e-suas-provas\/","title":{"rendered":"Leis de DeMorgan, Distribui\u00e7\u00e3o e Suas Provas"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Leis de DeMorgan, Distribui\u00e7\u00e3o e Suas Provas<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>Nesta aula, revisamos as provas das leis de DeMorgan de Distribui\u00e7\u00e3o para conjun\u00e7\u00e3o e disjun\u00e7\u00e3o, que s\u00e3o frequentemente usadas na l\u00f3gica proposicional e em \u00e1reas como teoria dos conjuntos, probabilidade, topologia, eletr\u00f4nica e programa\u00e7\u00e3o. S\u00e3o apresentadas as equival\u00eancias que formalizam a distribui\u00e7\u00e3o das nega\u00e7\u00f5es com conjun\u00e7\u00e3o e disjun\u00e7\u00e3o, bem como as regras de distributividade entre conjun\u00e7\u00e3o e disjun\u00e7\u00e3o. S\u00e3o explicadas as t\u00e9cnicas de dedu\u00e7\u00e3o utilizadas para obter essas provas, e os alunos s\u00e3o incentivados a completar as provas propostas para refor\u00e7ar seus conhecimentos. O exerc\u00edcio de perguntar a si mesmo: \u00abPosso construir essas provas em uma ordem diferente seguindo essa mesma metodologia?\u00bb tamb\u00e9m \u00e9 sugerido para melhorar as habilidades em l\u00f3gica.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de\n<\/p>\n<ol>\n<li><strong>Provar<\/strong> as leis de DeMorgan e as regras de distributividade entre conjun\u00e7\u00e3o e disjun\u00e7\u00e3o.<\/li>\n<li><strong>Aplicar<\/strong> as t\u00e9cnicas de dedu\u00e7\u00e3o aprendidas para a prova das leis de DeMorgan e distributividade.<\/li>\n<li><strong>Comparar<\/strong> as provas das leis de DeMorgan e distributividade para encontrar semelhan\u00e7as e diferen\u00e7as.<\/li>\n<li><strong>Analisar<\/strong> as provas das leis de DeMorgan e distributividade para melhorar a compreens\u00e3o da l\u00f3gica proposicional.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\">LEIS DE DEMORGAN<\/a><br \/>\n<a href=\"#2\">REGRAS DE DISTRIBUTIVIDADE ENTRE CONJUN\u00c7\u00c3O E DISJUN\u00c7\u00c3O<\/a><br \/>\n<a href=\"#3\">CONSIDERA\u00c7\u00d5ES FINAIS<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ntfTrdqIipo\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Agora \u00e9 hora de revisar outra propriedade frequentemente usada na l\u00f3gica proposicional, nomeadamente as provas das leis de DeMorgan de Distribui\u00e7\u00e3o para conjun\u00e7\u00e3o e disjun\u00e7\u00e3o. O uso dessas leis \u00e9 comum na teoria dos conjuntos e, por extens\u00e3o, permeia toda a matem\u00e1tica: desde a teoria das probabilidades, topologia e at\u00e9 tem presen\u00e7a na eletr\u00f4nica e na programa\u00e7\u00e3o. Como de costume, vamos detalhar as provas dessas leis usando as t\u00e9cnicas de dedu\u00e7\u00e3o que obtivemos at\u00e9 agora.<\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Leis de DeMorgan<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ntfTrdqIipo&amp;t=709s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">As leis de DeMorgan<\/span><\/strong><\/a> s\u00e3o um conjunto de equival\u00eancias que formalizam a distribui\u00e7\u00e3o das nega\u00e7\u00f5es com conjun\u00e7\u00e3o e disjun\u00e7\u00e3o. Formalmente, s\u00e3o expressas atrav\u00e9s das equival\u00eancias:<\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha \\wedge \\beta) \\dashv \\vdash (\\neg\\alpha \\vee \\neg \\beta)<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha \\vee \\beta) \\dashv \\vdash (\\neg\\alpha \\wedge \\neg \\beta)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Essas equival\u00eancias provadas podem ser obtidas sem a necessidade de fazer uma prova como a que fizemos at\u00e9 agora, pois podemos nos valer das defini\u00e7\u00f5es que relacionam conjun\u00e7\u00f5es com disjun\u00e7\u00f5es e um pouco de jogo com a equival\u00eancia da dupla nega\u00e7\u00e3o e substitui\u00e7\u00f5es. Da defini\u00e7\u00e3o de conjun\u00e7\u00e3o, segue-se que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A \\wedge B):= \\neg(\\neg A \\vee \\neg B)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Aplicando uma nega\u00e7\u00e3o a ambos os lados dessa express\u00e3o, temos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A \\wedge B):= \\neg\\neg(\\neg A \\vee \\neg B)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Ent\u00e3o, pela equival\u00eancia da dupla nega\u00e7\u00e3o, obtemos<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A \\wedge B)\\dashv \\vdash (\\neg A \\vee \\neg B)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, substituindo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\alpha<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B=\\beta<\/span><\/span>, obtemos a primeira equival\u00eancia de DeMorgan<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\neg(\\alpha \\wedge \\beta) \\dashv \\vdash (\\neg\\alpha \\vee \\neg \\beta)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Para obter a segunda, podemos continuar jogando com a express\u00e3o que t\u00ednhamos antes de fazer a substitui\u00e7\u00e3o, adicionando uma nega\u00e7\u00e3o a ambos os lados novamente, obtendo<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg(A \\wedge B)\\dashv \\vdash \\neg(\\neg A \\vee \\neg B)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">E ent\u00e3o, pela dupla nega\u00e7\u00e3o, obtemos<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg A \\vee \\neg B) \\dashv \\vdash (A \\wedge B)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se nesta \u00faltima express\u00e3o substituirmos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\neg\\alpha<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B=\\neg\\beta<\/span><\/span>, chegaremos a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg \\neg\\alpha \\vee \\neg \\neg\\beta) \\dashv \\vdash (\\neg\\alpha \\wedge \\neg\\beta)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Que, devido \u00e0 equival\u00eancia da dupla nega\u00e7\u00e3o, conduzir\u00e1 \u00e0 segunda equival\u00eancia de DeMorgan<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\neg( \\alpha \\vee \\beta) \\dashv \\vdash (\\neg\\alpha \\wedge \\neg\\beta)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Al\u00e9m disso, de forma completamente an\u00e1loga, podemos obter algumas formas adicionais, que s\u00e3o apenas varia\u00e7\u00f5es das que acabamos de revisar<\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg\\alpha \\wedge \\beta) \\dashv \\vdash (\\alpha \\vee \\neg \\beta)<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg\\alpha \\vee \\beta) \\dashv \\vdash (\\alpha \\wedge \\neg \\beta)<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha \\wedge \\neg\\beta) \\dashv \\vdash (\\neg\\alpha \\vee \\beta)<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha \\vee \\neg\\beta) \\dashv \\vdash (\\neg\\alpha \\wedge \\beta)<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Regras de Distributividade Entre Conjun\u00e7\u00e3o e Disjun\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ntfTrdqIipo&amp;t=709s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Como o nome sugere<\/span><\/strong><\/a>, essas regras nos permitem distribuir os conjuntos e disjun\u00e7\u00f5es dentro de uma express\u00e3o. Essas leis s\u00e3o resumidas nas seguintes duas equival\u00eancias:<\/p>\n<table style=\"text-align: left; color: #000000;\">\n<tbody>\n<tr>\n<td>\u2227 &#8211; Distributividade<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge(\\beta \\vee \\gamma)) \\dashv \\vdash ((\\alpha \\wedge \\beta)\\vee(\\alpha \\wedge \\gamma)) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u2228 &#8211; Distributividade<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee(\\beta \\wedge \\gamma)) \\dashv \\vdash ((\\alpha \\vee \\beta)\\wedge(\\alpha \\vee \\gamma)) <\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Como j\u00e1 vimos at\u00e9 agora, embora este seja um resultado conhecido, sua prova n\u00e3o \u00e9 nada trivial. Embora, para completar essa prova, deve-se raciocinar em ambas as dire\u00e7\u00f5es, desta vez darei a prova em apenas um sentido; a prova no sentido reverso ser\u00e1 deixada como um exerc\u00edcio para o leitor.<\/p>\n<h3>\u2227 &#8211; Distributividade<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ntfTrdqIipo&amp;t=831s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Para provar que<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma))\\}\\vdash((\\alpha \\wedge \\beta)\\vee(\\alpha \\wedge \\gamma))<\/span><\/span> ocorre, temos o seguinte racioc\u00ednio.<\/p>\n<table style=\"text-align: left; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\beta \\}\\vdash (\\alpha \\wedge(\\beta \\vee\\gamma)) <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\beta \\}\\vdash \\alpha <\/span><\/span><\/td>\n<td>; \u2227-elimina\u00e7\u00e3o(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\beta \\}\\vdash \\beta <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\beta \\}\\vdash (\\alpha\\wedge \\beta) <\/span><\/span><\/td>\n<td>; \u2227-Introdu\u00e7\u00e3o(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\beta \\}\\vdash ((\\alpha\\wedge \\beta)\\vee(\\alpha \\wedge \\gamma) )<\/span><\/span><\/td>\n<td>; \u2228-Introdu\u00e7\u00e3o(4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash (\\alpha \\wedge(\\beta \\vee\\gamma)) <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash (\\beta \\vee\\gamma) <\/span><\/span><\/td>\n<td>; \u2227-Elimina\u00e7\u00e3o(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash\\neg\\beta <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash\\gamma <\/span><\/span><\/td>\n<td>; \u2228-Elimina\u00e7\u00e3o(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash\\alpha <\/span><\/span><\/td>\n<td>; \u2227-Elimina\u00e7\u00e3o(6)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash (\\alpha\\wedge\\gamma) <\/span><\/span><\/td>\n<td>; \u2227-Introdu\u00e7\u00e3o(9,10)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma)), \\neg\\beta \\}\\vdash ((\\alpha\\wedge\\beta)\\vee(\\alpha\\wedge\\gamma)) <\/span><\/span><\/td>\n<td>; \u2228-Introdu\u00e7\u00e3o(11)<\/td>\n<\/tr>\n<tr>\n<td>(13)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\wedge(\\beta \\vee\\gamma))\\}\\vdash ((\\alpha\\wedge\\beta)\\vee(\\alpha\\wedge\\gamma))} <\/span><\/span><\/td>\n<td>; Casos(5,12)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Com isso, fica provado que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge(\\beta \\vee\\gamma))\\}\\vdash((\\alpha \\wedge \\beta)\\vee(\\alpha \\wedge \\gamma))<\/span><\/span>. Agora \u00e9 a sua vez de testar o que aprendeu e se aventurar a provar por conta pr\u00f3pria que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{((\\alpha \\wedge \\beta)\\vee(\\alpha \\wedge \\gamma))\\}\\vdash (\\alpha \\wedge(\\beta \\vee\\gamma))<\/span><\/span>.<\/p>\n<h3>\u2228 &#8211; Distributividade<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ntfTrdqIipo&amp;t=1449s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A prova de<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash((\\alpha \\vee \\beta)\\wedge(\\alpha \\vee \\gamma))<\/span><\/span> \u00e9 obtida a partir do seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: left; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma)), \\neg\\alpha\\}\\vdash (\\alpha \\vee(\\beta \\wedge\\gamma))<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma)), \\neg\\alpha\\}\\vdash \\neg\\alpha<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma)), \\neg\\alpha\\}\\vdash (\\beta \\wedge\\gamma)<\/span><\/span><\/td>\n<td>; \u2228-Elimina\u00e7\u00e3o(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma)), \\neg\\alpha\\}\\vdash \\beta<\/span><\/span><\/td>\n<td>; \u2227-Elimina\u00e7\u00e3o(3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma)), \\neg\\alpha\\}\\vdash \\gamma<\/span><\/span><\/td>\n<td>; \u2227-Elimina\u00e7\u00e3o(3)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash (\\neg\\alpha\\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; TD(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash (\\alpha\\vee \\beta)<\/span><\/span><\/td>\n<td>; <span class=\"katex-eq\" data-katex-display=\"false\">\\rightarrow<\/span>-Defini\u00e7\u00e3o(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash (\\neg\\alpha \\rightarrow \\gamma)<\/span><\/span><\/td>\n<td>; TD(5)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash (\\alpha \\vee \\gamma)<\/span><\/span><\/td>\n<td>; <span class=\"katex-eq\" data-katex-display=\"false\">\\rightarrow<\/span>-Defini\u00e7\u00e3o(8)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\vee(\\beta \\wedge\\gamma))\\}\\vdash ((\\alpha\\vee \\beta) \\wedge (\\alpha \\vee \\gamma))}<\/span><\/span><\/td>\n<td>; \u2227-Introdu\u00e7\u00e3o(7,9)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Esta \u00e9 metade da prova; agora, falta a prova reversa, mas isso fica como exerc\u00edcio para o leitor :3<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Considera\u00e7\u00f5es Finais<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Com esta revis\u00e3o das provas das leis de De Morgan de distribui\u00e7\u00e3o de conjun\u00e7\u00e3o e disjun\u00e7\u00e3o, podemos concluir nosso estudo sobre t\u00e9cnicas de dedu\u00e7\u00e3o na l\u00f3gica proposicional e como elas convergem na prova das leis da l\u00f3gica cl\u00e1ssica, ou pelo menos as mais importantes.<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c9 importante completar todas as provas propostas para refor\u00e7ar o conhecimento dessas t\u00e9cnicas. Para tornar isso um pouco menos complicado, \u00e9 muito conveniente comparar as provas em busca de semelhan\u00e7as, pois \u00e9 poss\u00edvel que a estrat\u00e9gia que funcionou em uma prova funcione com algumas varia\u00e7\u00f5es para alcan\u00e7ar outra.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Uma \u00faltima coisa que vale a pena notar \u00e9 a ordem que escolhi para desenvolver essas provas. Voc\u00ea deve notar que cada prova usou os resultados de algumas das provas anteriores. Escolhi essa ordem porque pessoalmente achei mais f\u00e1cil dessa forma. Um bom exerc\u00edcio para melhorar suas habilidades nessas coisas \u00e9 perguntar a si mesmo: \u00abPosso construir essas provas em uma ordem diferente seguindo essa mesma metodologia?\u00bb Recomendo fortemente que voc\u00ea tente obter essas provas em uma ordem diferente e use cada prova para obter as seguintes, porque, mesmo que voc\u00ea n\u00e3o consiga, a pr\u00e1tica que surge da tentativa lhe dar\u00e1 uma melhor compreens\u00e3o das provas e dos m\u00e9todos usados na l\u00f3gica.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Leis de DeMorgan, Distribui\u00e7\u00e3o e Suas Provas RESUMONesta aula, revisamos as provas das leis de DeMorgan de Distribui\u00e7\u00e3o para conjun\u00e7\u00e3o e disjun\u00e7\u00e3o, que s\u00e3o frequentemente usadas na l\u00f3gica proposicional e em \u00e1reas como teoria dos conjuntos, probabilidade, topologia, eletr\u00f4nica e programa\u00e7\u00e3o. S\u00e3o apresentadas as equival\u00eancias que formalizam a distribui\u00e7\u00e3o das nega\u00e7\u00f5es com conjun\u00e7\u00e3o e disjun\u00e7\u00e3o, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27484,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":22,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-27490","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-matematica-pt","category-logica-proposicional-pt","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>Leis de DeMorgan, Distribui\u00e7\u00e3o e Suas Provas - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra as Leis de DeMorgan e as regras de distributividade na l\u00f3gica proposicional. 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