{"id":28306,"date":"2021-04-18T13:00:14","date_gmt":"2021-04-18T13:00:14","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28306"},"modified":"2024-09-04T19:16:40","modified_gmt":"2024-09-04T19:16:40","slug":"normal-forms-and-their-properties","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/normal-forms-and-their-properties\/","title":{"rendered":"Normal Forms and Their Properties"},"content":{"rendered":"<p><center><\/p>\n<h1>Normal Forms and Their Properties<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARY<\/strong><br \/><em>Propositional logic is a fundamental tool in mathematics and computer science. In this class, an interesting and useful result related to normal forms will be presented. To achieve this, the concepts of literal, conjunctive normal form (CNF), and disjunctive normal form (DNF) will be defined. Moreover, the normal forms theorem will be demonstrated, which states that all propositional logic expressions are equivalent to an expression in DNF and another in CNF. The proof will be carried out by induction on the complexity of formulas, establishing that all propositional logic expressions can be written in DNF and CNF. This class will be very useful for understanding the foundations of propositional logic and applying them in various areas of knowledge.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Recall<\/strong> the definition of literal and of conjunctive and disjunctive normal forms.<\/li>\n<li><strong>Identify<\/strong> the structures of an expression in CNF and DNF.<\/li>\n<li><strong>Use<\/strong> CNF or DNF to simplify propositional logic expressions.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">DEFINITION OF LITERAL<\/a><br \/>\n<a href=\"#2\">DEFINITION OF NORMAL FORMS<\/a><br \/>\n<a href=\"#3\">NORMAL FORMS THEOREM<\/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;\">An interesting and useful result of propositional logic is related to normal forms. To delve into these topics, we first need to review some concepts.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Definition of 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>A literal is any<\/strong><\/a> atomic expression or the negation of an atomic expression. Based on this, we talk about negative or positive literals depending on whether the atomic expressions are preceded by a negation or not. For example: <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> would be a positive literal and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg A<\/span><\/span> would be a negative literal.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Definition of Normal Forms<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=337s\" target=\"_blank\" rel=\"noopener\"><strong>An expression <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> is in normal form<\/strong><\/a> when it can be written as a conjunction of disjunctions of literals, that is:<\/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;\">Similarly, there will be a disjunctive normal form (DNF) if it is written as the disjunction of conjunctions of literals:<\/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>Normal Forms Theorem<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=446s\" target=\"_blank\" rel=\"noopener\"><strong>All propositional logic expressions<\/strong><\/a> are equivalent to an expression in DNF and another in CNF.<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000080;\"><strong>PROOF:<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\">This can be demonstrated by induction on the complexity of formulas <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Base case:<\/strong> If <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> is an atomic expression, it can be written in CNF and DNF simultaneously because: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv F_D \\equiv F_C<\/span><\/span>, where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_C:=((F\\vee F)\\wedge (F\\vee F)) <\/span><\/span> and <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>Inductive step:<\/strong> Let <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> be any two expressions for which the theorem holds; that is, there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_C<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_C<\/span><\/span> in CNF, and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_D<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_D<\/span><\/span> in DNF such that:\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>    So we can write:<\/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>    Without loss of generality, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:= \\neg G<\/span><\/span>, then using the <strong>substitution theorem<\/strong> on the <strong>generalized De Morgan&#8217;s laws<\/strong>, we have that:<\/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>    On the other hand, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=G\\wedge H<\/span><\/span>, then by the substitution theorem:<\/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>    which is a conjunctive normal form. And in the same way, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=H\\vee G,<\/span><\/span> then:<\/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>    that is, a disjunctive normal form.<\/p>\n<p>    Therefore, the induction is complete and all propositional logic expressions can be written in DNF and CNF.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">The study of conjunctive normal form (CNF) and disjunctive normal form (DNF) in propositional logic is essential for simplifying and solving complex problems in mathematics and computer science. The theorem that states that any logical expression can be written in both DNF and CNF is highly relevant, as it allows for structuring propositions in a more manageable and standardized way, thus facilitating their analysis and manipulation. The importance of this result lies in providing a solid foundation for algorithm design, logical expression optimization, and efficient problem-solving in various fields of knowledge, such as artificial intelligence and software verification. Additionally, the proof technique by induction used to prove this theorem strengthens the understanding of the fundamental properties of logical expressions and their applicability in other mathematical contexts.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Normal Forms and Their Properties SUMMARYPropositional logic is a fundamental tool in mathematics and computer science. In this class, an interesting and useful result related to normal forms will be presented. To achieve this, the concepts of literal, conjunctive normal form (CNF), and disjunctive normal form (DNF) will be defined. Moreover, the normal forms theorem [&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":12,"footnotes":""},"categories":[605,567,619],"tags":[],"class_list":["post-28306","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematical-logic","category-mathematics","category-propositional-logic"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Normal Forms and Their Properties - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"A normal form is a standardized logical structure that simplifies propositional expressions in mathematics and computer science.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/normal-forms-and-their-properties\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Normal Forms and Their Properties\" \/>\n<meta property=\"og:description\" content=\"A normal form is a standardized logical structure that simplifies propositional expressions in mathematics and computer science.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/normal-forms-and-their-properties\/\" \/>\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=\"2021-04-18T13:00:14+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-04T19:16:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/04\/formanormal.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1792\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Normal Forms and Their Properties\" \/>\n<meta name=\"twitter:description\" content=\"A normal form is a standardized logical structure that simplifies propositional expressions in mathematics and computer science.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/04\/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\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Normal Forms and Their Properties\",\"datePublished\":\"2021-04-18T13:00:14+00:00\",\"dateModified\":\"2024-09-04T19:16:40+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/\"},\"wordCount\":916,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/formanormal.jpg\",\"articleSection\":[\"Mathematical Logic\",\"Mathematics\",\"Propositional Logic\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/normal-forms-and-their-properties\\\/\",\"name\":\"Normal Forms and Their Properties - 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