{"id":28297,"date":"2021-04-17T13:00:26","date_gmt":"2021-04-17T13:00:26","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28297"},"modified":"2024-09-04T18:09:16","modified_gmt":"2024-09-04T18:09:16","slug":"induction-sur-la-complexite-des-formules","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/","title":{"rendered":"Induction sur la complexit\u00e9 des formules"},"content":{"rendered":"<p><center><\/p>\n<h1>Induction sur la complexit\u00e9 des formules<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00c9SUM\u00c9<\/strong><br \/><em>Dans ce cours, vous apprendrez une variante de l&#8217;induction math\u00e9matique, connue sous le nom d'\u00bbinduction sur la complexit\u00e9 des expressions\u00bb, qui est tr\u00e8s utile pour d\u00e9montrer des propri\u00e9t\u00e9s en logique propositionnelle. \u00c0 travers un exemple simple, le th\u00e9or\u00e8me de substitution, vous verrez comment cette technique est appliqu\u00e9e et comment elle peut prouver qu&#8217;une propri\u00e9t\u00e9 s&#8217;applique \u00e0 toutes les expressions de la logique propositionnelle. De plus, il sera expliqu\u00e9 comment fonctionne l&#8217;hypoth\u00e8se d&#8217;induction et le pas inductif, afin que vous puissiez appliquer cette technique dans vos propres d\u00e9monstrations.<\/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>Comprendre<\/strong> le concept d&#8217;induction sur la complexit\u00e9 des expressions.<\/li>\n<li><strong>Appliquer<\/strong> l&#8217;induction math\u00e9matique sur la complexit\u00e9 des formules en logique propositionnelle.<\/li>\n<li><strong>Comprendre<\/strong> la d\u00e9monstration par induction sur la complexit\u00e9 et comment elle est appliqu\u00e9e en logique propositionnelle.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>CONTENU<\/strong><br \/>\n<a href=\"#1\">INDUCTION SUR LA COMPLEXIT\u00c9<\/a><br \/>\n<a href=\"#2\">UN EXEMPLE SIMPLE : LE TH\u00c9OR\u00c8ME DE SUBSTITUTION<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/wipHo3D2lQY\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Induction sur la complexit\u00e9<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=407s\" target=\"_blank\" rel=\"noopener\"><strong>Supposons que nous voulions prouver qu&#8217;une propri\u00e9t\u00e9<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> s&#8217;applique \u00e0 toute expression <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>. Une mani\u00e8re de le d\u00e9montrer consiste \u00e0 utiliser la variante de l&#8217;<strong>induction math\u00e9matique<\/strong> appel\u00e9e \u00abinduction sur la complexit\u00e9 des expressions\u00bb. Cela se fait \u00e0 travers les \u00e9tapes suivantes :<\/p>\n<ul style=\"text-align: justify;\">\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=432s\" target=\"_blank\" rel=\"noopener\"><strong>Premi\u00e8rement, nous montrons que<\/strong><\/a> toutes les expressions atomiques satisfont cette propri\u00e9t\u00e9 (cela correspond au cas n=1 d&#8217;une induction traditionnelle).<\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong>Ensuite, en supposant que<\/strong><\/a> cette propri\u00e9t\u00e9 est vraie pour des expressions quelconques <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, nous prouvons qu&#8217;elle est \u00e9galement vraie pour des expressions de la forme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>; ou \u00e9quivalemment, pour <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> et l&#8217;une des suivantes : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G.<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Si nous r\u00e9ussissons cela, alors nous concluons que la propri\u00e9t\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> s&#8217;applique \u00e0 toutes les expressions de la logique propositionnelle. C&#8217;est ce qu&#8217;on appelle \u00abl&#8217;induction math\u00e9matique sur la complexit\u00e9 des expressions\u00bb.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Un exemple simple : le th\u00e9or\u00e8me de substitution<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=681s\" target=\"_blank\" rel=\"noopener\"><strong>Pour mieux comprendre comment fonctionne l&#8217;induction<\/strong><\/a> sur la complexit\u00e9 des formules, nous allons examiner le (m\u00e9ta)th\u00e9or\u00e8me de substitution.<\/p>\n<p style=\"text-align: justify;\">Supposons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Soit <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> une expression contenant <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> comme sous-expression, et soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> l&#8217;expression obtenue en rempla\u00e7ant toutes les occurrences de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<h3>D\u00e9monstration par induction sur la complexit\u00e9 des formules<\/h3>\n<p style=\"text-align: justify;\">La d\u00e9monstration par induction sur la complexit\u00e9 consiste \u00e0 montrer deux choses : 1) un cas initial (pour les formules atomiques) et 2) le pas inductif (si cela fonctionne pour des expressions quelconques <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, alors cela fonctionne pour <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>, ou plus simplement : cela fonctionne pour <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> et pour au moins une des suivantes : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G<\/span><\/span> ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\leftrightarrow G<\/span><\/span>).<\/p>\n<p style=\"text-align: justify;\">Supposons que <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> soit une expression atomique, <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est une sous-expression de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est le r\u00e9sultat du remplacement de toutes les sous-expressions <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, alors puisque <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> est atomique, nous avons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime \\equiv G<\/span><\/span>. D&#8217;autre part, comme <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> est atomique et <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est une sous-expression de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F<\/span><\/span>. Enfin, nous obtenons :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F \\equiv G \\equiv H^\\prime<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ainsi, le cas initial pour les expressions atomiques est d\u00e9montr\u00e9.<\/p>\n<p style=\"text-align: justify;\">Passons maintenant au pas inductif.<\/p>\n<h4>L&#8217;hypoth\u00e8se d&#8217;induction<\/h4>\n<p style=\"text-align: justify;\">Supposons que le th\u00e9or\u00e8me fonctionne pour deux expressions quelconques <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span><\/span>, chacune contenant <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> comme sous-expression et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F \\equiv G.<\/span><\/span> Alors si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1^\\prime<\/span><\/span> est ce que nous obtenons en rempla\u00e7ant toutes les <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> dans <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span><\/span> et que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2^\\prime<\/span><\/span> est ce que nous obtenons en rempla\u00e7ant toutes les <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> dans <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span><\/span>, alors nous avons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1\\equiv H_1^\\prime<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2\\equiv H_2^\\prime<\/span><\/span>.<\/p>\n<h4>Le pas inductif<\/h4>\n<p style=\"text-align: justify;\">Ici, nous allons v\u00e9rifier si, en cons\u00e9quence de l&#8217;hypoth\u00e8se d&#8217;induction, le th\u00e9or\u00e8me s&#8217;applique \u00e9galement \u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_1<\/span><\/span> (ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_2,<\/span><\/span> l&#8217;une ou l&#8217;autre) et pour au moins une des suivantes : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\wedge H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\vee H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\rightarrow H_2.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= \\neg H_1<\/span><\/span>, alors, selon l&#8217;hypoth\u00e8se d&#8217;induction, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv \\neg H_1^\\prime=: H^\\prime <\/span><\/span>, o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est le r\u00e9sultat du remplacement de toutes les occurrences de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> dans <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Par cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">De m\u00eame, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= H_1 \\wedge H_2<\/span><\/span>, alors, selon l&#8217;hypoth\u00e8se d&#8217;induction, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H_1^\\prime \\wedge H_2 \\equiv H_1^\\prime \\wedge H_2^\\prime =: H^\\prime <\/span><\/span>, o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est le r\u00e9sultat du remplacement de toutes les occurrences de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> dans <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Par cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ainsi, l&#8217;induction est compl\u00e8te et le th\u00e9or\u00e8me de substitution est valable pour toutes les expressions de la logique propositionnelle.<\/p>\n<p style=\"text-align: justify;\">\nEn appliquant cette forme d&#8217;induction, il est possible de garantir qu&#8217;une propri\u00e9t\u00e9 s&#8217;applique \u00e0 toutes les expressions d&#8217;un syst\u00e8me logique, ce qui est particuli\u00e8rement utile en logique propositionnelle pour structurer des d\u00e9monstrations rigoureuses. De plus, son utilit\u00e9 s&#8217;\u00e9tend \u00e0 des domaines tels que l&#8217;intelligence artificielle et le d\u00e9veloppement de logiciels, o\u00f9 la v\u00e9rification des syst\u00e8mes logiques est essentielle. Gr\u00e2ce \u00e0 cette technique, il est possible d&#8217;automatiser les d\u00e9monstrations et d&#8217;assurer la coh\u00e9rence des expressions complexes, ce qui r\u00e9duit le risque d&#8217;erreurs et am\u00e9liore la pr\u00e9cision dans les environnements d\u00e9pendant de la validit\u00e9 math\u00e9matique et logique.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Induction sur la complexit\u00e9 des formules R\u00c9SUM\u00c9Dans ce cours, vous apprendrez une variante de l&#8217;induction math\u00e9matique, connue sous le nom d&#8217;\u00bbinduction sur la complexit\u00e9 des expressions\u00bb, qui est tr\u00e8s utile pour d\u00e9montrer des propri\u00e9t\u00e9s en logique propositionnelle. \u00c0 travers un exemple simple, le th\u00e9or\u00e8me de substitution, vous verrez comment cette technique est appliqu\u00e9e et comment [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28280,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[617,631,569],"tags":[],"class_list":["post-28297","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>Induction sur la complexit\u00e9 des formules - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"La inducci\u00f3n sobre la complejidad es una t\u00e9cnica matem\u00e1tica utilizada para demostrar propiedades generales de f\u00f3rmulas y expresiones l\u00f3gicas.\" \/>\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\/induction-sur-la-complexite-des-formules\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Induction sur la complexit\u00e9 des formules\" \/>\n<meta property=\"og:description\" content=\"L&#039;induction sur la complexit\u00e9 est une technique math\u00e9matique utilis\u00e9e pour d\u00e9montrer les propri\u00e9t\u00e9s g\u00e9n\u00e9rales des formules et des expressions logiques.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\" \/>\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=\"2021-04-17T13:00:26+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-04T18:09:16+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Induction sur la complexit\u00e9 des formules\" \/>\n<meta name=\"twitter:description\" content=\"L&#039;induction sur la complexit\u00e9 est une technique math\u00e9matique utilis\u00e9e pour d\u00e9montrer les propri\u00e9t\u00e9s g\u00e9n\u00e9rales des formules et des expressions logiques.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.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\/induction-sur-la-complexite-des-formules\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Induction sur la complexit\u00e9 des formules\",\"datePublished\":\"2021-04-17T13:00:26+00:00\",\"dateModified\":\"2024-09-04T18:09:16+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\"},\"wordCount\":1090,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg\",\"articleSection\":[\"Logique Math\u00e9matique\",\"Logique Propositionnelle\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\",\"url\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\",\"name\":\"Induction sur la complexit\u00e9 des formules - toposuranos.com\/material\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg\",\"datePublished\":\"2021-04-17T13:00:26+00:00\",\"dateModified\":\"2024-09-04T18:09:16+00:00\",\"description\":\"La inducci\u00f3n sobre la complejidad es una t\u00e9cnica matem\u00e1tica utilizada para demostrar propiedades generales de f\u00f3rmulas y expresiones l\u00f3gicas.\",\"breadcrumb\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Induction sur la complexit\u00e9 des formules\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\/\/toposuranos.com\/material\/#website\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"name\":\"toposuranos.com\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\/\/toposuranos.com\/material\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\",\"name\":\"toposuranos.com\/material\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\/material\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/groups\/toposuranos\",\"https:\/\/x.com\/topuranos\",\"https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\/\/www.linkedin.com\/company\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\/\/toposuranos.com\/material\"],\"url\":\"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Induction sur la complexit\u00e9 des formules - toposuranos.com\/material","description":"La inducci\u00f3n sobre la complejidad es una t\u00e9cnica matem\u00e1tica utilizada para demostrar propiedades generales de f\u00f3rmulas y expresiones l\u00f3gicas.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/","og_locale":"es_ES","og_type":"article","og_title":"Induction sur la complexit\u00e9 des formules","og_description":"L'induction sur la complexit\u00e9 est une technique math\u00e9matique utilis\u00e9e pour d\u00e9montrer les propri\u00e9t\u00e9s g\u00e9n\u00e9rales des formules et des expressions logiques.","og_url":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-04-17T13:00:26+00:00","article_modified_time":"2024-09-04T18:09:16+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad-1024x585.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Induction sur la complexit\u00e9 des formules","twitter_description":"L'induction sur la complexit\u00e9 est une technique math\u00e9matique utilis\u00e9e pour d\u00e9montrer les propri\u00e9t\u00e9s g\u00e9n\u00e9rales des formules et des expressions logiques.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Induction sur la complexit\u00e9 des formules","datePublished":"2021-04-17T13:00:26+00:00","dateModified":"2024-09-04T18:09:16+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/"},"wordCount":1090,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg","articleSection":["Logique Math\u00e9matique","Logique Propositionnelle","Math\u00e9matiques"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/","url":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/","name":"Induction sur la complexit\u00e9 des formules - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg","datePublished":"2021-04-17T13:00:26+00:00","dateModified":"2024-09-04T18:09:16+00:00","description":"La inducci\u00f3n sobre la complejidad es una t\u00e9cnica matem\u00e1tica utilizada para demostrar propiedades generales de f\u00f3rmulas y expresiones l\u00f3gicas.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.jpg","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/fr\/induction-sur-la-complexite-des-formules\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Induction sur la complexit\u00e9 des formules"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/28297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=28297"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/28297\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/28280"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=28297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=28297"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=28297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}