{"id":27507,"date":"2021-02-14T13:00:26","date_gmt":"2021-02-14T13:00:26","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27507"},"modified":"2024-07-21T16:39:20","modified_gmt":"2024-07-21T16:39:20","slug":"lois-de-demorgan-de-distribution-et-leurs-demonstrations-2","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/lois-de-demorgan-de-distribution-et-leurs-demonstrations-2\/","title":{"rendered":"Lois de DeMorgan, de Distribution et leurs d\u00e9monstrations"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Lois de DeMorgan, de Distribution et leurs d\u00e9monstrations<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00c9SUM\u00c9<\/strong><br \/><em>Dans cette le\u00e7on, nous examinons les d\u00e9monstrations des lois de DeMorgan de Distribution de la conjonction et de la disjonction, qui sont fr\u00e9quemment utilis\u00e9es dans la logique propositionnelle et dans des domaines tels que la th\u00e9orie des ensembles, les probabilit\u00e9s, la topologie, l&#8217;\u00e9lectronique et la programmation. Les \u00e9quivalences qui formalisent la distribution des n\u00e9gations avec la conjonction et la disjonction, ainsi que les r\u00e8gles de distributivit\u00e9 entre la conjonction et la disjonction, sont pr\u00e9sent\u00e9es. Les techniques de d\u00e9duction utilis\u00e9es pour obtenir ces d\u00e9monstrations sont expliqu\u00e9es et les \u00e9tudiants sont encourag\u00e9s \u00e0 compl\u00e9ter les d\u00e9monstrations propos\u00e9es pour renforcer leurs connaissances. Il est \u00e9galement sugg\u00e9r\u00e9 de se poser la question \u00abPourrai-je construire ces d\u00e9monstrations dans un ordre diff\u00e9rent en suivant cette m\u00eame m\u00e9thodologie?\u00bb pour am\u00e9liorer les comp\u00e9tences en logique.<\/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 cette le\u00e7on, l&#8217;\u00e9tudiant sera capable de\n<\/p>\n<ol>\n<li><strong>D\u00e9montrer<\/strong> les lois de DeMorgan et les r\u00e8gles de distributivit\u00e9 entre la conjonction et la disjonction.<\/li>\n<li><strong>Appliquer<\/strong> les techniques de d\u00e9duction apprises pour la d\u00e9monstration des lois de DeMorgan et de distributivit\u00e9.<\/li>\n<li><strong>Comparer<\/strong> les d\u00e9monstrations des lois de DeMorgan et de distributivit\u00e9 pour en rechercher les similitudes et les diff\u00e9rences.<\/li>\n<li><strong>Analyser<\/strong> les d\u00e9monstrations des lois de DeMorgan et de distributivit\u00e9 pour am\u00e9liorer la compr\u00e9hension de la logique propositionnelle.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">LOIS DE DEMORGAN<\/a><br \/>\n<a href=\"#2\">R\u00c8GLES DE DISTRIBUTIVIT\u00c9 ENTRE LA CONJONCTION ET LA DISJONCTION<\/a><br \/>\n<a href=\"#3\">CONSID\u00c9RATIONS FINALES<\/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;\">Il est maintenant temps d&#8217;examiner une autre des propri\u00e9t\u00e9s fr\u00e9quemment utilis\u00e9es dans la logique propositionnelle, \u00e0 savoir les d\u00e9monstrations des lois de DeMorgan de Distribution de la conjonction et de la disjonction. L&#8217;utilisation de ces lois est courante en ce qui concerne la th\u00e9orie des ensembles et, par extension, elles impr\u00e8gnent toute la math\u00e9matique : de la th\u00e9orie des probabilit\u00e9s, la topologie et m\u00eame l&#8217;\u00e9lectronique et la programmation. Comme d&#8217;habitude, nous examinerons les d\u00e9monstrations de ces lois en utilisant les techniques de d\u00e9duction que nous avons obtenues jusqu&#8217;\u00e0 pr\u00e9sent.<\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Lois 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;\">Les lois de DeMorgan<\/span><\/strong><\/a> sont un ensemble d&#8217;\u00e9quivalences qui formalisent la distribution des n\u00e9gations avec la conjonction et la disjonction. Formellement, elles s&#8217;expriment par les \u00e9quivalences suivantes :<\/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;\">Ces \u00e9quivalences prouv\u00e9es peuvent \u00eatre obtenues sans la n\u00e9cessit\u00e9 de faire une d\u00e9monstration comme celle que nous avons faite jusqu&#8217;\u00e0 pr\u00e9sent, car nous pouvons nous appuyer sur les d\u00e9finitions qui relient les conjonctions avec les disjonctions et un peu de jeu avec l&#8217;\u00e9quivalence de la double n\u00e9gation et les substitutions. De la d\u00e9finition de la conjonction, il s&#8217;ensuit 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;\">En appliquant une n\u00e9gation aux deux c\u00f4t\u00e9s de cette expression, nous avons 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;\">Ensuite, par l&#8217;\u00e9quivalence de la double n\u00e9gation, nous obtenons<\/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;\">Enfin, en rempla\u00e7ant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A = \\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B = \\beta<\/span><\/span>, nous obtenons la premi\u00e8re \u00e9quivalence 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;\">Pour obtenir la deuxi\u00e8me, nous pouvons continuer \u00e0 jouer avec l&#8217;expression que nous avions avant de faire le remplacement en ajoutant \u00e0 nouveau une n\u00e9gation aux deux c\u00f4t\u00e9s, obtenant<\/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;\">Et ensuite, par double n\u00e9gation, nous obtenons que<\/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;\">Si nous rempla\u00e7ons dans cette derni\u00e8re expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A = \\neg\\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B = \\neg\\beta<\/span><\/span>, nous arrivons \u00e0<\/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;\">Ce qui, en raison de l&#8217;\u00e9quivalence de la double n\u00e9gation, conduira \u00e0 la deuxi\u00e8me \u00e9quivalence 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;\">De plus, de mani\u00e8re tout \u00e0 fait analogue, nous pouvons obtenir certaines formes suppl\u00e9mentaires, qui ne sont que des variations de celles que nous venons de revoir<\/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>R\u00e8gles de Distributivit\u00e9 entre la Conjonction et la Disjonction<\/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;\">Comme son nom l&#8217;indique<\/span><\/strong><\/a>, ces r\u00e8gles nous permettent de distribuer les ensembles et les disjonctions dans une expression. Ces lois se r\u00e9sument dans les deux \u00e9quivalences suivantes :<\/p>\n<table style=\"text-align: left; color: #000000;\">\n<tbody>\n<tr>\n<td>\u2227 &#8211; Distributivit\u00e9<\/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; Distributivit\u00e9<\/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;\">Comme nous l&#8217;avons d\u00e9j\u00e0 vu jusqu&#8217;\u00e0 pr\u00e9sent, bien qu&#8217;il s&#8217;agisse d&#8217;un r\u00e9sultat connu, sa d\u00e9monstration n&#8217;a rien de trivial. Bien que, pour compl\u00e9ter cette d\u00e9monstration, il faille raisonner dans les deux sens, cette fois, je ne donnerai la d\u00e9monstration que dans un seul sens, la d\u00e9monstration dans le sens inverse restera comme un exercice pour le lecteur.<\/p>\n<h3>\u2227 &#8211; Distributivit\u00e9<\/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;\">Pour d\u00e9montrer 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>, le raisonnement suivant est utilis\u00e9 :<\/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>; Pr\u00e9misse<\/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-\u00e9limination(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>; Pr\u00e9misse<\/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-Introduction(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-Introduction(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>; Pr\u00e9misse<\/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-\u00e9limination(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>; Pr\u00e9misse<\/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-\u00e9limination(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-\u00e9limination(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-Introduction(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-Introduction(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>; Cas(5,12)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Avec cela, il est d\u00e9montr\u00e9 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>. Maintenant, c&#8217;est \u00e0 vous de mettre \u00e0 l&#8217;\u00e9preuve ce que vous avez appris et de vous aventurer \u00e0 d\u00e9montrer par vous-m\u00eame 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; Distributivit\u00e9<\/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;\">La d\u00e9monstration 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> est obtenue \u00e0 partir du raisonnement suivant :<\/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>; Pr\u00e9misse<\/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>; Pr\u00e9misse<\/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-\u00e9limination(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-\u00e9limination(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-\u00e9limination(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>-D\u00e9finition(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>-D\u00e9finition(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-Introduction(7,9)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Ceci est la moiti\u00e9 de la d\u00e9monstration, il reste maintenant \u00e0 la faire en sens inverse, mais cela reste comme exercice pour le lecteur :3<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Consid\u00e9rations Finales<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Avec cette r\u00e9vision que nous avons faite des d\u00e9monstrations des lois de De Morgan de distribution de la conjonction et de la disjonction, nous pouvons conclure notre \u00e9tude sur les techniques de d\u00e9duction de la logique propositionnelle et comment elles convergent dans la d\u00e9monstration des lois de la logique classique, ou du moins les plus importantes.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Il est important de compl\u00e9ter toutes les d\u00e9monstrations propos\u00e9es pour renforcer les connaissances sur ces techniques. Pour rendre cela un peu moins compliqu\u00e9, il est tr\u00e8s utile de comparer les d\u00e9monstrations \u00e0 la recherche de similitudes, car il est possible que la strat\u00e9gie qui a fonctionn\u00e9 dans une d\u00e9monstration fonctionne avec quelques variations pour en r\u00e9aliser une autre.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Une derni\u00e8re chose qui vaut la peine d&#8217;\u00eatre not\u00e9e est l&#8217;ordre que j&#8217;ai choisi pour d\u00e9velopper ces d\u00e9monstrations. Vous devez noter que chaque d\u00e9monstration a utilis\u00e9 les r\u00e9sultats de certaines des d\u00e9monstrations pr\u00e9c\u00e9dentes. J&#8217;ai choisi cet ordre car, personnellement, je l&#8217;ai trouv\u00e9 plus simple de cette mani\u00e8re. Un bon exercice pour am\u00e9liorer vos comp\u00e9tences dans ces domaines est de vous poser la question \u00abPourrai-je construire ces d\u00e9monstrations dans un ordre diff\u00e9rent en suivant cette m\u00eame m\u00e9thodologie?\u00bb. Je vous recommande vivement d&#8217;essayer d&#8217;obtenir ces d\u00e9monstrations dans un ordre diff\u00e9rent et d&#8217;utiliser chaque d\u00e9monstration pour obtenir les suivantes, car m\u00eame si vous n&#8217;y parvenez pas, la pratique qui d\u00e9coule de la tentative vous apportera une meilleure compr\u00e9hension des d\u00e9monstrations et des m\u00e9thodes utilis\u00e9es en logique.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lois de DeMorgan, de Distribution et leurs d\u00e9monstrations R\u00c9SUM\u00c9Dans cette le\u00e7on, nous examinons les d\u00e9monstrations des lois de DeMorgan de Distribution de la conjonction et de la disjonction, qui sont fr\u00e9quemment utilis\u00e9es dans la logique propositionnelle et dans des domaines tels que la th\u00e9orie des ensembles, les probabilit\u00e9s, la topologie, l&#8217;\u00e9lectronique et la programmation. Les [&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":57,"footnotes":""},"categories":[617,631,569],"tags":[],"class_list":["post-27507","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Lois de DeMorgan, de Distribution et leurs d\u00e9monstrations - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les lois de DeMorgan et les r\u00e8gles de distributivit\u00e9 en logique propositionnelle. 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