{"id":27409,"date":"2021-01-28T13:00:51","date_gmt":"2021-01-28T13:00:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27409"},"modified":"2024-07-09T02:09:33","modified_gmt":"2024-07-09T02:09:33","slug":"5-simetrias-de-la-logica-proposicional","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/","title":{"rendered":"5 Simetr\u00edas de la L\u00f3gica Proposicional"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>5 Simetr\u00edas de la L\u00f3gica Proposicional<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Resumen:<\/strong><br \/>A lo largo de esta clase, exploraremos c\u00f3mo la doble negaci\u00f3n, el silogismo hipot\u00e9tico, el contrapositivo de la implicancia, los teoremas de deducci\u00f3n y las definiciones de los conectores se combinan para formar las simetr\u00edas de la l\u00f3gica proposicional. Con demostraciones claras y sencillas, aprender\u00e1s a dominar las equivalencias y a aplicarlas en tus propios desaf\u00edos l\u00f3gicos.<\/p>\n<p>Las simetr\u00edas abordadas en la clase incluyen: <span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span>-Simetr\u00eda, <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-Simetr\u00eda, <span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span>-Simetr\u00eda, <span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span>-Simetr\u00eda y <span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span>-Simetr\u00eda. Adem\u00e1s, se destacan las interacciones entre las demostraciones y c\u00f3mo cada una se apoya en las anteriores para simplificar futuras deducciones. Esta clase no solo te proporcionar\u00e1 un conocimiento profundo de la L\u00f3gica Proposicional, sino que tambi\u00e9n te ense\u00f1ar\u00e1 a utilizar demostraciones previas para optimizar tu proceso de aprendizaje.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Objetivos de Aprendizaje<\/u>:<\/strong><br \/>Al finalizar esta clase el estudiante ser\u00e1 capaz de<\/p>\n<ol>\n<li><strong>Recordar<\/strong> conceptos b\u00e1sicos de la l\u00f3gica proposicional, como el silogismo hipot\u00e9tico y la doble negaci\u00f3n.<\/li>\n<li><strong>Reconocer<\/strong> las 5 Simetr\u00edas de la L\u00f3gica Proposicional.<\/li>\n<li><strong>Comprender<\/strong> el proceso de demostraci\u00f3n de las equivalencias de simetr\u00edas.<\/li>\n<li><strong>Aplicar<\/strong> la Presunci\u00f3n, el Teorema de Deducci\u00f3n y su Rec\u00edproco en las demostraciones.<\/li>\n<li><strong>Relacionar<\/strong> las definiciones de conectores l\u00f3gicos con las simetr\u00edas.<\/li>\n<li><strong>Valorar<\/strong> la importancia de realizar demostraciones una sola vez y reutilizarlas en futuras demostraciones.<\/li>\n<li><strong>Desarrollar<\/strong> habilidades anal\u00edticas y cr\u00edticas al realizar demostraciones l\u00f3gicas.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDICE DE CONTENIDOS<\/strong><br \/>\n<a href=\"#1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span> &#8211; SIMETR\u00cdA<\/a><br \/>\n<a href=\"#2\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span> &#8211; SIMETR\u00cdA<\/a><br \/>\n<a href=\"#3\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span> &#8211; SIMETR\u00cdA<\/a><br \/>\n<a href=\"#4\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span> &#8211; SIMETR\u00cdA<\/a><br \/>\n<a href=\"#5\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span> &#8211; SIMETR\u00cdA<\/a><br \/>\n<a href=\"#6\">OBSERVACIONES FINALES<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/XKY_nS5Mizk\" 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;\">Una consecuencia directa del silogismo hipot\u00e9tico, la doble negaci\u00f3n y el contrapositivo de la implicancia, los teoremas de deducci\u00f3n y las definiciones de los conectores son las 5 simetr\u00edas de la l\u00f3gica proposicional que revisaremos a continuaci\u00f3n.<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\downarrow \\beta) \\dashv\\vdash (\\beta\\downarrow \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta) \\dashv\\vdash (\\beta\\vee \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta) \\dashv\\vdash (\\beta\\wedge \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta) \\dashv\\vdash (\\beta\\leftrightarrow \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta) \\dashv\\vdash (\\beta\\veebar\\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Las demostraciones de estas equivalencias no son del todo triviales, pero a diferencia de algunas demostraciones que ya hemos visto, s\u00ed son bastante sencillas. A continuaci\u00f3n, se muestra la demostraci\u00f3n de cada una en una sola direcci\u00f3n; la demostraci\u00f3n en el sentido inverso es pr\u00e1cticamente id\u00e9ntica y queda como ejercicio para el lector realizarlas.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Simetr\u00eda<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=203s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">ver<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta)\\}\\vdash (\\alpha \\vee\\beta)<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta)\\}\\vdash (\\neg \\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta) := (\\neg \\alpha \\rightarrow \\beta)<\/span><\/span><\/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)\\}\\vdash (\\neg \\beta \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; CPI(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\vee \\beta)\\}\\vdash ( \\beta \\vee \\alpha)}<\/span><\/span><\/td>\n<td>; porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( \\beta \\vee \\alpha) := (\\neg\\beta\\rightarrow\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">El razonamiento en el sentido inverso se obtiene con muy pocas variaciones iniciando con la presunci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\beta\\vee\\alpha)\\}\\vdash (\\beta\\vee\\alpha)<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-Simetr\u00eda<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=383s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">ver<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash \\neg(\\alpha \\downarrow \\beta) <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash (\\alpha \\vee \\beta) <\/span><\/span><\/td>\n<td>; de (1) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\vee\\beta) := \\neg(\\alpha \\downarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash (\\beta \\vee \\alpha) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Simetr\u00eda<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\alpha \\downarrow \\beta) \\rightarrow (\\beta \\vee \\alpha)) <\/span><\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\beta \\vee \\alpha) \\rightarrow (\\alpha \\downarrow \\beta)) <\/span><\/span><\/td>\n<td>; CPI(4)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\beta \\downarrow \\alpha) \\rightarrow (\\alpha \\downarrow \\beta)) <\/span><\/span><\/td>\n<td>; de (5) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta\\vee\\alpha) := \\neg(\\beta \\downarrow \\alpha)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\beta \\downarrow \\alpha) \\} \\vdash (\\alpha \\downarrow \\beta)}<\/span><\/span><\/td>\n<td>; RTD(6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, razonando en el sentido inverso se obtiene la deducci\u00f3n en el sentido contrario iniciando con la presunci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg(\\beta\\downarrow\\alpha)\\}\\vdash \\neg(\\beta\\downarrow\\alpha)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-Simetr\u00eda<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=659s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">ver<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\neg\\alpha \\downarrow \\neg\\beta)<\/span><\/span><\/td>\n<td>; de (1) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta) := (\\neg\\alpha \\downarrow \\neg\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\neg\\beta \\downarrow \\neg\\alpha)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-Simetr\u00eda (2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\wedge \\beta)\\} \\vdash ( \\beta \\wedge \\alpha)}<\/span><\/span><\/td>\n<td>; de (3) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta \\wedge \\alpha) := (\\neg\\beta \\downarrow \\neg\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Igual que en anterior, si razonas en el sentido inverso se obtiene con muy pocas variaciones iniciando con la presunci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{( \\beta \\wedge \\alpha)\\}\\vdash ( \\beta \\wedge \\alpha)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-Simetr\u00eda<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=830s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">ver<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ((\\alpha \\rightarrow \\beta) \\wedge (\\alpha \\rightarrow \\beta))<\/span><\/span><\/td>\n<td>; De (1) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta) := ((\\alpha \\rightarrow \\beta) \\wedge (\\beta \\rightarrow \\alpha)) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ((\\beta \\rightarrow \\alpha) \\wedge (\\alpha \\rightarrow \\beta) )<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-Simetr\u00eda(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\beta \\leftrightarrow \\alpha)}<\/span><\/span><\/td>\n<td>; De (3) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta \\leftrightarrow \\alpha) := ((\\beta \\rightarrow \\alpha) \\wedge (\\alpha \\rightarrow \\beta)) <\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Igual que el anterior, pero iniciando con la presunci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{( \\beta \\leftrightarrow \\alpha)\\}\\vdash ( \\beta\\leftrightarrow \\alpha)<\/span><\/span><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span><\/span>-Simetr\u00eda<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=982s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">ver<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\alpha \\leftrightarrow \\beta) <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ( \\beta \\leftrightarrow \\alpha) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-Simetr\u00eda(1)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\alpha \\leftrightarrow \\beta) \\rightarrow ( \\beta \\leftrightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg ( \\beta \\leftrightarrow \\alpha) \\rightarrow \\neg (\\alpha \\leftrightarrow \\beta)) <\/span><\/span><\/td>\n<td>; CPI(3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg ( \\beta \\leftrightarrow \\alpha)\\} \\vdash \\neg (\\alpha \\leftrightarrow \\beta) <\/span><\/span><\/td>\n<td>; RTD(4)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{ ( \\beta \\veebar \\alpha)\\} \\vdash (\\alpha \\veebar \\beta)} <\/span><\/span><\/td>\n<td>; De (5) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( \\beta \\veebar \\alpha) := \\neg\\beta \\leftrightarrow \\alpha)<\/span><\/span> y <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta) := \\neg (\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Como en todos los dem\u00e1s casos, basta con probar la presunci\u00f3n en sentido inverso <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\beta \\leftrightarrow \\alpha)\\} \\vdash (\\beta \\leftrightarrow \\alpha)<\/span><\/span> para obtener la deducci\u00f3n en dicho sentido.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Observaciones Finales<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=1145s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Un aspecto al que el lector debe prestar atenci\u00f3n<\/span><\/strong><\/a> es al orden en el que se han elegido demostrar estas 5 simetr\u00edas de la l\u00f3gica proposicional. N\u00f3tese que cada una est\u00e1 hecha de tal modo que utiliza alguna de las demostraciones previamente realizadas. Esto refleja el enfoque que se debe seguir al realizar demostraciones: \u00e9stas se llevan a cabo una sola vez (\u00a1y nunca m\u00e1s!); despu\u00e9s de eso, tu objetivo debe centrarse en utilizar las demostraciones anteriores para simplificar las futuras.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>5 Simetr\u00edas de la L\u00f3gica Proposicional Resumen:A lo largo de esta clase, exploraremos c\u00f3mo la doble negaci\u00f3n, el silogismo hipot\u00e9tico, el contrapositivo de la implicancia, los teoremas de deducci\u00f3n y las definiciones de los conectores se combinan para formar las simetr\u00edas de la l\u00f3gica proposicional. Con demostraciones claras y sencillas, aprender\u00e1s a dominar las equivalencias [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27410,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[600,602,563],"tags":[],"class_list":["post-27409","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-matematica","category-logica-proposicional","category-matematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>5 Simetr\u00edas de la L\u00f3gica Proposicional - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubre las 5 Simetr\u00edas de la L\u00f3gica Proposicional con demostraciones claras: doble negaci\u00f3n, silogismo hipot\u00e9tico, y m\u00e1s.\" \/>\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\/es\/5-simetrias-de-la-logica-proposicional\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"5 Simetr\u00edas de la L\u00f3gica Proposicional\" \/>\n<meta property=\"og:description\" content=\"Descubre las 5 Simetr\u00edas de la L\u00f3gica Proposicional con demostraciones claras: doble negaci\u00f3n, silogismo hipot\u00e9tico, y m\u00e1s.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/\" \/>\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=\"2021-01-28T13:00:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-07-09T02:09:33+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"356\" \/>\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=\"5 Simetr\u00edas de la L\u00f3gica Proposicional\" \/>\n<meta name=\"twitter:description\" content=\"Descubre las 5 Simetr\u00edas de la L\u00f3gica Proposicional con demostraciones claras: doble negaci\u00f3n, silogismo hipot\u00e9tico, y m\u00e1s.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.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=\"5 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"5 Simetr\u00edas de la L\u00f3gica Proposicional\",\"datePublished\":\"2021-01-28T13:00:51+00:00\",\"dateModified\":\"2024-07-09T02:09:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/\"},\"wordCount\":1197,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg\",\"articleSection\":[\"L\u00f3gica Matem\u00e1tica\",\"L\u00f3gica Proposicional\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/\",\"url\":\"http:\/\/toposuranos.com\/material\/es\/5-simetrias-de-la-logica-proposicional\/\",\"name\":\"5 Simetr\u00edas de la L\u00f3gica Proposicional - 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