{"id":28310,"date":"2021-04-18T13:00:20","date_gmt":"2021-04-18T13:00:20","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28310"},"modified":"2024-09-04T19:16:55","modified_gmt":"2024-09-04T19:16:55","slug":"formas-normais-e-suas-propriedades","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/formas-normais-e-suas-propriedades\/","title":{"rendered":"Formas Normais e suas Propriedades"},"content":{"rendered":"<p><center><\/p>\n<h1>Formas Normais e suas Propriedades<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>A l\u00f3gica proposicional \u00e9 uma ferramenta fundamental na matem\u00e1tica e na inform\u00e1tica. Nesta aula, ser\u00e1 apresentado um resultado interessante e \u00fatil relacionado \u00e0s formas normais. Para isso, ser\u00e3o definidos os conceitos de literal, forma normal conjuntiva (FNC) e forma normal disjuntiva (FND). Al\u00e9m disso, ser\u00e1 demonstrado o teorema das formas normais, que estabelece que todas as express\u00f5es da l\u00f3gica proposicional s\u00e3o equivalentes a uma express\u00e3o em FND e outra em FNC. A demonstra\u00e7\u00e3o ser\u00e1 feita por indu\u00e7\u00e3o sobre a complexidade das f\u00f3rmulas, estabelecendo que todas as express\u00f5es da l\u00f3gica proposicional podem ser escritas em FND e FNC. Esta aula ser\u00e1 muito \u00fatil para entender os fundamentos da l\u00f3gica proposicional e aplic\u00e1-los em diversas \u00e1reas do conhecimento.<\/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 estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Recordar<\/strong> a defini\u00e7\u00e3o de literal e das formas normais conjuntivas e disjuntivas.<\/li>\n<li><strong>Identificar<\/strong> as estruturas de uma express\u00e3o em FNC e FND.<\/li>\n<li><strong>Utilizar<\/strong> FNC ou FND para simplificar express\u00f5es de l\u00f3gica proposicional.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\">DEFINI\u00c7\u00c3O DE LITERAL<\/a><br \/>\n<a href=\"#2\">DEFINI\u00c7\u00c3O DE FORMAS NORMAIS<\/a><br \/>\n<a href=\"#3\">TEOREMA DAS FORMAS NORMAIS<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CrTcmmE4Q6c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">Um resultado interessante e \u00fatil da l\u00f3gica proposicional est\u00e1 relacionado com as formas normais. Para detalhar esses t\u00f3picos, primeiro precisamos revisar alguns conceitos.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Defini\u00e7\u00e3o de Literal<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=309s\" target=\"_blank\" rel=\"noopener\"><strong>Um literal \u00e9 qualquer<\/strong><\/a> express\u00e3o at\u00f4mica ou a nega\u00e7\u00e3o de uma express\u00e3o at\u00f4mica. Com base nisso, falamos de literais negativos ou positivos, dependendo se as express\u00f5es at\u00f4micas s\u00e3o precedidas ou n\u00e3o por uma nega\u00e7\u00e3o. Por exemplo: <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> seria um literal positivo e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg A<\/span><\/span> seria um literal negativo.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Defini\u00e7\u00e3o de Formas Normais<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=337s\" target=\"_blank\" rel=\"noopener\"><strong>Uma express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est\u00e1 em forma normal<\/strong><\/a> conjuntiva (FNC) quando pode ser escrita como uma conjun\u00e7\u00e3o de disjun\u00e7\u00f5es de literais, ou seja:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F=\\bigwedge_{i=1}^n \\left( \\bigvee_{j=1}^m L_{ij}\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">E, de maneira semelhante, haver\u00e1 uma forma normal disjuntiva (FND) se for escrita como a disjun\u00e7\u00e3o de conjun\u00e7\u00f5es de literais:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F=\\bigvee_{i=1}^n \\left(\\bigwedge_{j=1}^m L_{ij}\\right)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Teorema das Formas Normais<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=446s\" target=\"_blank\" rel=\"noopener\"><strong>Todas as express\u00f5es da l\u00f3gica proposicional<\/strong><\/a> s\u00e3o equivalentes a uma express\u00e3o em FND e outra em FNC.<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\">Isso pode ser demonstrado por indu\u00e7\u00e3o sobre a complexidade das f\u00f3rmulas <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Caso base:<\/strong> Se <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u00e9 uma express\u00e3o at\u00f4mica, pode ser escrita em FNC e FND ao mesmo tempo, porque: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv F_D \\equiv F_C<\/span><\/span>, onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_C:=((F\\vee F)\\wedge (F\\vee F)) <\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_D:=((F\\wedge F)\\vee (F\\wedge F)) <\/span><\/span><\/li>\n<li><strong>Passo indutivo:<\/strong> Sejam <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> duas express\u00f5es quaisquer para as quais o resultado do teorema se aplica; ou seja, existem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_C<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_C<\/span><\/span> em FNC, e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_D<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_D<\/span><\/span> em FND, tais que:\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\equiv G_D \\equiv G_D<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H_D \\equiv H_D<\/span><\/span><\/p>\n<p>    Assim podemos escrever:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle G_D := \\bigvee_{i=1}^n \\bigwedge_{j=1}^m L_{ij}^{GD}<\/span><\/span> ; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle G_C := \\bigwedge_{i=1}^n \\bigvee_{j=1}^m L_{ij}^{GC}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle H_D := \\bigvee_{i=1}^n \\bigwedge_{j=1}^m L_{ij}^{HD}<\/span><\/span> ; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle H_C := \\bigwedge_{i=1}^n \\bigvee_{j=1}^m L_{ij}^{HC}<\/span><\/span><\/p>\n<p>    Sem perda de generalidade, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:= \\neg G<\/span><\/span>, ent\u00e3o usando o <strong>teorema de substitui\u00e7\u00e3o<\/strong> sobre as <strong>leis generalizadas de De Morgan<\/strong>, teremos que:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F:= \\neg G \\equiv \\left\\{ \\begin{matrix}\n\n    \\neg G_D := \\neg \\bigvee_{i=1}^n \\bigwedge_{j=1}^m L_{ij}^{GD} \\equiv\\bigwedge_{i=1}^n \\neg \\bigwedge_{j=1}^m L_{ij}^{GD} \\equiv \\bigwedge_{i=1}^n \\bigvee_{j=1}^m \\neg L_{ij}^{GD} \\\\ \\\\ \\neg G_C := \\neg \\bigwedge_{i=1}^n \\bigvee_{j=1}^m L_{ij}^{GC} \\equiv \\bigvee_{i=1}^n \\neg \\bigvee_{j=1}^m L_{ij}^{GC} \\equiv \\bigvee_{i=1}^n \\bigwedge_{j=1}^m \\neg L_{ij}^{GC} \\end{matrix}\\right. <\/span>\n<p>    Por outro lado, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=G\\wedge H<\/span><\/span>, ent\u00e3o pelo teorema de substitui\u00e7\u00e3o:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F:=G\\wedge H \\equiv G_C \\wedge H_C := \\bigwedge_{i=1}^n \\bigvee_{j=1}^m L_{ij}^{GC} \\wedge \\bigwedge_{i=1}^{n^\\prime} \\bigvee_{j=1}^{m^\\prime} L_{ij}^{HC} <\/span><\/span><\/p>\n<p>    que \u00e9 uma forma normal conjuntiva. E de modo an\u00e1logo, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=H\\vee G,<\/span><\/span> ent\u00e3o:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F:=G\\wedge H \\equiv G_D \\vee H_D := \\bigvee_{i=1}^n \\bigwedge_{j=1}^m L_{ij}^{GD} \\vee \\bigvee_{i=1}^{\\overline{n}} \\bigwedge_{j=1}^{\\overline{m}} L_{ij}^{HD} <\/span><\/span><\/p>\n<p>    ou seja, uma forma normal disjuntiva.<\/p>\n<p>    Portanto, a indu\u00e7\u00e3o est\u00e1 completa e todas as express\u00f5es da l\u00f3gica proposicional podem ser escritas em FND e FNC.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">O estudo das formas normais conjuntiva (FNC) e disjuntiva (FND) da l\u00f3gica proposicional \u00e9 fundamental para a simplifica\u00e7\u00e3o e resolu\u00e7\u00e3o de problemas complexos em matem\u00e1tica e inform\u00e1tica. O teorema que estabelece que qualquer express\u00e3o l\u00f3gica pode ser escrita tanto em FND quanto em FNC \u00e9 de grande relev\u00e2ncia, pois permite estruturar as proposi\u00e7\u00f5es de forma mais gerenci\u00e1vel e padronizada, facilitando sua an\u00e1lise e manipula\u00e7\u00e3o. A import\u00e2ncia desse resultado reside em fornecer uma base s\u00f3lida para o design de algoritmos, a otimiza\u00e7\u00e3o de express\u00f5es l\u00f3gicas e a resolu\u00e7\u00e3o eficiente de problemas em diversas \u00e1reas do conhecimento, como intelig\u00eancia artificial e verifica\u00e7\u00e3o de software. Al\u00e9m disso, a t\u00e9cnica de demonstra\u00e7\u00e3o por indu\u00e7\u00e3o utilizada para provar esse teorema refor\u00e7a a compreens\u00e3o das propriedades fundamentais das express\u00f5es l\u00f3gicas e sua aplicabilidade em outros contextos matem\u00e1ticos.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Formas Normais e suas Propriedades RESUMOA l\u00f3gica proposicional \u00e9 uma ferramenta fundamental na matem\u00e1tica e na inform\u00e1tica. Nesta aula, ser\u00e1 apresentado um resultado interessante e \u00fatil relacionado \u00e0s formas normais. Para isso, ser\u00e3o definidos os conceitos de literal, forma normal conjuntiva (FNC) e forma normal disjuntiva (FND). Al\u00e9m disso, ser\u00e1 demonstrado o teorema das formas [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28305,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":16,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-28310","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>Formas Normais e suas Propriedades - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Uma forma normal \u00e9 uma estrutura l\u00f3gica padronizada que simplifica as express\u00f5es proposicionais em matem\u00e1tica e inform\u00e1tica.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/pt\/formas-normais-e-suas-propriedades\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formas Normais e suas Propriedades\" \/>\n<meta property=\"og:description\" content=\"Uma forma normal \u00e9 uma estrutura l\u00f3gica padronizada que simplifica as express\u00f5es proposicionais em matem\u00e1tica e inform\u00e1tica.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/pt\/formas-normais-e-suas-propriedades\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-18T13:00:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-04T19:16:55+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/formanormal-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Formas Normais e suas Propriedades\" \/>\n<meta name=\"twitter:description\" content=\"Uma forma normal \u00e9 uma estrutura l\u00f3gica padronizada que simplifica as express\u00f5es proposicionais em matem\u00e1tica e inform\u00e1tica.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/formanormal.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Formas Normais e suas Propriedades\",\"datePublished\":\"2021-04-18T13:00:20+00:00\",\"dateModified\":\"2024-09-04T19:16:55+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/\"},\"wordCount\":1002,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/formanormal.jpg\",\"articleSection\":[\"L\u00f3gica Matem\u00e1tica\",\"L\u00f3gica Proposicional\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/formas-normais-e-suas-propriedades\\\/\",\"name\":\"Formas Normais e suas Propriedades - 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