{"id":247,"date":"2015-02-24T09:02:41","date_gmt":"2015-02-24T15:02:41","guid":{"rendered":"http:\/\/www.uv.mx\/personal\/aguerra\/?p=247"},"modified":"2015-02-24T09:02:41","modified_gmt":"2015-02-24T15:02:41","slug":"ejercicios-ia2-deduccion-natural","status":"publish","type":"post","link":"https:\/\/www.uv.mx\/personal\/aguerra\/2015\/02\/24\/ejercicios-ia2-deduccion-natural\/","title":{"rendered":"Ejercicios IA2: Deducci\u00f3n natural"},"content":{"rendered":"
    \n
  1. Demuestre que el siguiente argumento es v\u00e1lido: \u03d5<\/span>\u2192<\/span>\u03c8<\/span>,<\/span>\u03c8<\/span>\u2192<\/span>\u03c1<\/span>\u22a2<\/span>\u03d5<\/span>\u2192<\/span>\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span>\n

    Escribimos las premisas y la conclusi\u00f3n<\/p>\n\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    2<\/td>\n\u03c8<\/span>\u2192<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    …<\/td>\n…<\/td>\n…<\/td>\n<\/tr>\n
    n<\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\nConclusi\u00f3n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Como la estructura de la conclusi\u00f3n es una implicaci\u00f3n, adoptamos su antecedente como supuesto y tratamos de demostrar su consecuente.<\/p>\n\n\n\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    2<\/td>\n\u03c8<\/span>\u2192<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    3<\/td>\n\u03d5<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    …<\/td>\n…<\/td>\n…<\/td>\n<\/tr>\n
    n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/td>\n\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\nConclusi\u00f3n<\/td>\n<\/tr>\n
    n<\/span><\/span><\/span><\/span><\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    El resto es sencillo, dos modus ponens<\/em> y una introducci\u00f3n de conjunci\u00f3n:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    2<\/td>\n\u03c8<\/span>\u2192<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\nPremisa<\/td>\n<\/tr>\n
    3<\/td>\n\u03d5<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    4<\/td>\n\u03c8<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>e<\/span>)<\/span>\u00a0<\/span>1<\/span>,<\/span>3<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    5<\/td>\n\u03c1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>e<\/span>)<\/span>\u00a0<\/span>2<\/span>,<\/span>4<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    6<\/td>\n\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2227<\/span>i<\/span>)<\/span>\u00a0<\/span>4<\/span>,<\/span>5<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    7<\/td>\n\u03d5<\/span>\u2192<\/span>\u03c8<\/span>\u2227<\/span>\u03c1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span>\u00a0<\/span>3<\/span>\u2212<\/span>6<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Recuerden que los pasos 3 a 6, forman una caja en los esquemas de las notas del curso. Se puede decir que es una demostraci\u00f3n dentro de la demostraci\u00f3n original, dado que \u03d5<\/span><\/span><\/span><\/span> es el caso.<\/li>\n

  2. Demostrar que la siguiente proposici\u00f3n es un teorema: (<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span>\u2192<\/span>(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span>.\n

    Como el operador principal de la f\u00f3rmula es una implicaci\u00f3n, procedemos a usar la introducci\u00f3n de la implicaci\u00f3n para la prueba:<\/p>\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n…<\/td>\n<\/tr>\n
    n<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span>\u2192<\/span>(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Como el operador principal del paso n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span> tambi\u00e9n es una implicaci\u00f3n, procedemos de la misma manera:<\/p>\n\n\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    2<\/td>\n\u03b1<\/span>\u2227<\/span>\u03b2<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/td>\n\u03b3<\/span><\/span><\/span><\/span><\/td>\n…<\/td>\n<\/tr>\n
    n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    n<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span>\u2192<\/span>(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Procedemos con la demostraci\u00f3n aprovechando la conjunci\u00f3n para probar las implicaciones:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
    Paso<\/th>\nF\u00f3rmula<\/th>\nRegla<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    2<\/td>\n\u03b1<\/span>\u2227<\/span>\u03b2<\/span><\/span><\/span><\/span><\/td>\nSupuesto<\/td>\n<\/tr>\n
    3<\/td>\n\u03b1<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2227<\/span>e<\/span>1<\/span>)<\/span>\u00a0<\/span>2<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    4<\/td>\n\u03b2<\/span>\u2192<\/span>\u03b3<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>e<\/span>)<\/span>\u00a0<\/span>1<\/span>,<\/span>3<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    5<\/td>\n\u03b2<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2227<\/span>e<\/span>2<\/span>)<\/span>\u00a0<\/span>2<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    6<\/td>\n\u03b3<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>e<\/span>)<\/span>\u00a0<\/span>4<\/span>,<\/span>5<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    7<\/td>\n(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span>\u00a0<\/span>2<\/span>\u2212<\/span>6<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    8<\/td>\n(<\/span>\u03b1<\/span>\u2192<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span>\u2192<\/span>(<\/span>\u03b1<\/span>\u2227<\/span>\u03b2<\/span>\u2192<\/span>\u03b3<\/span>)<\/span><\/span><\/span><\/span><\/td>\n(<\/span>\u2192<\/span>i<\/span>)<\/span>\u00a0<\/span>1<\/span>\u2212<\/span>7<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Demuestre que el siguiente argumento es v\u00e1lido: \u03d5\u2192\u03c8,\u03c8\u2192\u03c1\u22a2\u03d5\u2192\u03c8\u2227\u03c1 Escribimos las premisas y la conclusi\u00f3n Paso F\u00f3rmula Regla 1 \u03d5\u2192\u03c8 Premisa 2 \u03c8\u2192\u03c1 Premisa … … … n \u03d5\u2192\u03c8\u2227\u03c1 Conclusi\u00f3n Como la estructura de la conclusi\u00f3n es una implicaci\u00f3n, adoptamos su antecedente como supuesto y tratamos de demostrar su consecuente. Paso F\u00f3rmula Regla 1 \u03d5\u2192\u03c8 Premisa…<\/p>\n","protected":false},"author":1701,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-247","post","type-post","status-publish","format-standard","hentry","category-miblog"],"_links":{"self":[{"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/posts\/247","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/users\/1701"}],"replies":[{"embeddable":true,"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/comments?post=247"}],"version-history":[{"count":0,"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/posts\/247\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/media?parent=247"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/categories?post=247"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.uv.mx\/personal\/aguerra\/wp-json\/wp\/v2\/tags?post=247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}