{"id":28323,"date":"2021-04-18T13:00:57","date_gmt":"2021-04-18T13:00:57","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28323"},"modified":"2024-09-05T01:48:12","modified_gmt":"2024-09-05T01:48:12","slug":"formes-normales-et-leurs-proprietes","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/","title":{"rendered":"Formes Normales et leurs Propri\u00e9t\u00e9s"},"content":{"rendered":"<p><center><\/p>\n<h1>Formes Normales et leurs Propri\u00e9t\u00e9s<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00c9SUM\u00c9<\/strong><br \/><em>La logique propositionnelle est un outil fondamental en math\u00e9matiques et en informatique. Dans ce cours, un r\u00e9sultat int\u00e9ressant et utile concernant les formes normales sera pr\u00e9sent\u00e9. Pour cela, les concepts de litt\u00e9ral, forme normale conjonctive (FNC) et forme normale disjonctive (FND) seront d\u00e9finis. De plus, le th\u00e9or\u00e8me des formes normales sera d\u00e9montr\u00e9, \u00e9tablissant que toutes les expressions de la logique propositionnelle sont \u00e9quivalentes \u00e0 une expression en FND et une autre en FNC. La d\u00e9monstration sera r\u00e9alis\u00e9e par induction sur la complexit\u00e9 des formules, ce qui permettra d&#8217;\u00e9tablir que toutes les expressions de la logique propositionnelle peuvent \u00eatre \u00e9crites en FND et FNC. Ce cours sera tr\u00e8s utile pour comprendre les bases de la logique propositionnelle et les appliquer dans divers domaines de la connaissance.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Se rappeler<\/strong> la d\u00e9finition de litt\u00e9ral et des formes normales conjonctives et disjonctives.<\/li>\n<li><strong>Identifier<\/strong> les structures d&#8217;une expression en FNC et FND.<\/li>\n<li><strong>Utiliser<\/strong> FNC ou FND pour simplifier les expressions de la logique propositionnelle.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">D\u00c9FINITION DU LITT\u00c9RAL<\/a><br \/>\n<a href=\"#2\">D\u00c9FINITION DES FORMES NORMALES<\/a><br \/>\n<a href=\"#3\">TH\u00c9OR\u00c8ME DES FORMES NORMALES<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CrTcmmE4Q6c\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">Un r\u00e9sultat int\u00e9ressant et utile de la logique propositionnelle est li\u00e9 aux formes normales. Pour aborder ces sujets en d\u00e9tail, il est n\u00e9cessaire de revoir d&#8217;abord certains concepts.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>D\u00e9finition du Litt\u00e9ral<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=309s\" target=\"_blank\" rel=\"noopener\"><strong>Un litt\u00e9ral est toute<\/strong><\/a> expression atomique ou la n\u00e9gation d&#8217;une expression atomique. En fonction de cela, on parle de litt\u00e9raux n\u00e9gatifs ou positifs selon que les expressions atomiques sont pr\u00e9c\u00e9d\u00e9es ou non d&#8217;une n\u00e9gation. Par exemple : <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> serait un litt\u00e9ral positif et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg A<\/span><\/span> serait un litt\u00e9ral n\u00e9gatif.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>D\u00e9finition des Formes Normales<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=337s\" target=\"_blank\" rel=\"noopener\"><strong>Une expression <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est en forme normale<\/strong><\/a> conjonctive (FNC) lorsqu&#8217;elle peut \u00eatre \u00e9crite comme une conjonction de disjonctions de litt\u00e9raux, c&#8217;est-\u00e0-dire :<\/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;\">De m\u00eame, on a une forme normale disjonctive (FND) si elle est \u00e9crite comme une disjonction de conjonctions de litt\u00e9raux :<\/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>Th\u00e9or\u00e8me des Formes Normales<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=CrTcmmE4Q6c&amp;t=446s\" target=\"_blank\" rel=\"noopener\"><strong>Toutes les expressions de la logique propositionnelle<\/strong><\/a> sont \u00e9quivalentes \u00e0 une expression en FND et une autre en FNC.<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\">Cela peut \u00eatre d\u00e9montr\u00e9 par induction sur la complexit\u00e9 des formules <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Cas de base :<\/strong> Si <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est une expression atomique, elle peut \u00eatre \u00e9crite \u00e0 la fois en FNC et en FND, car : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv F_D \\equiv F_C<\/span><\/span>, o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_C:=((F\\vee F)\\wedge (F\\vee F)) <\/span><\/span> et <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>\u00c9tape d&#8217;induction :<\/strong> Soient <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> deux expressions quelconques pour lesquelles le r\u00e9sultat du th\u00e9or\u00e8me est valable ; c&#8217;est-\u00e0-dire qu&#8217;il existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_C<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_C<\/span><\/span> en FNC, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_D<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G_D<\/span><\/span> en FND, tels 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>    Ainsi, nous pouvons \u00e9crire :<\/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>    Sans perte de g\u00e9n\u00e9ralit\u00e9, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:= \\neg G<\/span><\/span>, alors en utilisant le <strong>th\u00e9or\u00e8me de substitution<\/strong> et les <strong>lois g\u00e9n\u00e9ralis\u00e9es de De Morgan<\/strong>, on obtient :<\/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>    D&#8217;autre part, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=G\\wedge H<\/span><\/span>, alors selon le th\u00e9or\u00e8me de substitution :<\/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>    ce qui est une forme normale conjonctive. Et de m\u00eame, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F:=H\\vee G,<\/span><\/span> alors :<\/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>    c&#8217;est-\u00e0-dire une forme normale disjonctive.<\/p>\n<p>    Par cons\u00e9quent, l&#8217;induction est compl\u00e8te et toutes les expressions de la logique propositionnelle peuvent \u00eatre \u00e9crites en FND et FNC.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">L&#8217;\u00e9tude des formes normales conjonctive (FNC) et disjonctive (FND) de la logique propositionnelle est essentielle pour simplifier et r\u00e9soudre des probl\u00e8mes complexes en math\u00e9matiques et en informatique. Le th\u00e9or\u00e8me qui stipule que toute expression logique peut \u00eatre \u00e9crite \u00e0 la fois en FND et en FNC est d&#8217;une grande importance, car il permet de structurer les propositions de mani\u00e8re plus g\u00e9rable et standardis\u00e9e, facilitant ainsi leur analyse et manipulation. L&#8217;importance de ce r\u00e9sultat r\u00e9side dans le fait qu&#8217;il fournit une base solide pour la conception d&#8217;algorithmes, l&#8217;optimisation des expressions logiques et la r\u00e9solution efficace de probl\u00e8mes dans divers domaines de la connaissance, tels que l&#8217;intelligence artificielle et la v\u00e9rification des logiciels. De plus, la technique de preuve par induction utilis\u00e9e pour d\u00e9montrer ce th\u00e9or\u00e8me renforce la compr\u00e9hension des propri\u00e9t\u00e9s fondamentales des expressions logiques et de leur applicabilit\u00e9 dans d&#8217;autres contextes math\u00e9matiques.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Formes Normales et leurs Propri\u00e9t\u00e9s R\u00c9SUM\u00c9La logique propositionnelle est un outil fondamental en math\u00e9matiques et en informatique. Dans ce cours, un r\u00e9sultat int\u00e9ressant et utile concernant les formes normales sera pr\u00e9sent\u00e9. Pour cela, les concepts de litt\u00e9ral, forme normale conjonctive (FNC) et forme normale disjonctive (FND) seront d\u00e9finis. De plus, le th\u00e9or\u00e8me des formes normales [&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":14,"footnotes":""},"categories":[617,631,569],"tags":[],"class_list":["post-28323","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logique-mathematique","category-logique-propositionnelle","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Formes Normales et leurs Propri\u00e9t\u00e9s - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Une forme normale est une structure logique standardis\u00e9e qui simplifie les expressions propositionnelles en math\u00e9matiques et en informatique.\" \/>\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\/fr\/formes-normales-et-leurs-proprietes\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formes Normales et leurs Propri\u00e9t\u00e9s\" \/>\n<meta property=\"og:description\" content=\"Une forme normale est une structure logique standardis\u00e9e qui simplifie les expressions propositionnelles en math\u00e9matiques et en informatique.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/\" \/>\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=\"2021-04-18T13:00:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-05T01:48:12+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=\"Formes Normales et leurs Propri\u00e9t\u00e9s\" \/>\n<meta name=\"twitter:description\" content=\"Une forme normale est une structure logique standardis\u00e9e qui simplifie les expressions propositionnelles en math\u00e9matiques et en informatique.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Formes Normales et leurs Propri\u00e9t\u00e9s\",\"datePublished\":\"2021-04-18T13:00:57+00:00\",\"dateModified\":\"2024-09-05T01:48:12+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/\"},\"wordCount\":1057,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/formanormal.jpg\",\"articleSection\":[\"Logique Math\u00e9matique\",\"Logique Propositionnelle\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/\",\"url\":\"http:\/\/toposuranos.com\/material\/fr\/formes-normales-et-leurs-proprietes\/\",\"name\":\"Formes Normales et leurs Propri\u00e9t\u00e9s - 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