{"id":33901,"date":"2021-04-09T13:00:14","date_gmt":"2021-04-09T13:00:14","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33901"},"modified":"2025-08-01T02:15:01","modified_gmt":"2025-08-01T02:15:01","slug":"probationes-per-inductionem-generalizatio-de-morgan-et-distributio","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/probationes-per-inductionem-generalizatio-de-morgan-et-distributio\/","title":{"rendered":"Probationes per Inductionem: Generalizatio De Morgan et Distributio"},"content":{"rendered":"<p><center><\/p>\n<h1>Probationes per Inductionem: Regulae Generalizatae De Morgan et Distributio<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARIUM<\/strong><br \/><em>In hac lectione agitur de probationibus per inductionem in mathematica et logica propositionali. Exponuntur duo genera probationum: probationes internae sive deductivae, quae in regulis logicae nituntur, et probationes externae sive metamathematicae, quae necessariae sunt ad enuntiata probanda quae ad ipsam logicam spectant. Inductio Mathematica introducitur ut methodus demonstrationis, quae permittit demonstrare quaedam enuntiata valere pro omnibus numeris naturalibus. Exemplum cum demonstratione exhibetur, et exponuntur formae generalizatae legum De Morgan necnon legum distributivarum in logica propositionali, una cum earum demonstrationibus per inductionem. Haec lectio magni momenti est ad intellegendos fundamenta mathematicae et logicae, atque ad eos applicandos in variis cognitionis regionibus.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nPost hanc lectionem confectam, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Agnoscere<\/strong> duo genera probationum distinguenda: probationes internas sive deductivas et probationes externas sive metamathematicas.<\/li>\n<li><strong>Adhibere<\/strong> inductionem mathematicam ad demonstrationes faciendas de numeris naturalibus et in logica propositionali.<\/li>\n<li><strong>Uti<\/strong> notationibus conjunctionum et disjunctionum ad enuntiata logicae propositionalis scribenda.<\/li>\n<li><strong>Intellegere<\/strong> formas generalizatas legum De Morgan et Distributionis in Logica Propositionali.<\/li>\n<li><strong>Intellegere<\/strong> notionem hypotheseos inductionis eiusque munus in demonstratione per inductionem.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">PROBATIONES INTERNAE ET EXTERNAE<\/a><br \/>\n<a href=\"#2\">PROBATIONES PER INDUCTIONEM MATHEMATICAM<\/a><br \/>\n<a href=\"#3\">PROBATIONES PER INDUCTIONEM IN LOGICA PROPOSITIONALI<\/a><br \/>\n<a href=\"#4\">FORMA GENERALIZATA LEGUM DE MORGAN<\/a><br \/>\n<a href=\"#5\">FORMA GENERALIZATA LEGUM DISTRIBUTIVARUM<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/eJQcNPrKyW0\" title=\"YouTube video player\" 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>Probationes internae et externae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=eJQcNPrKyW0&amp;t=212s\" target=\"_blank\" rel=\"noopener\"><strong>Duo genera probationum distinguenda sunt.<\/strong><\/a> Hactenus multas probationes formales inspeximus. Huiusmodi probationes ex regulis logicae oriuntur. Taliae probationes dicuntur fieri \u00abintra logicam\u00bb, atque ideo eas etiam \u00abprobationes internas\u00bb sive deductivas appellamus. Huiusmodi probationes formales ambitum limitatum habent, quia solum valent ad enuntiata probanda quae in sermone logico scribi possunt. Tamen aliquando volumus probare quaedam de ipsa logica. Vellemus probare omnia enuntiata logicae propositionalis propriam aliquam qualitatem implere. Huiusmodi enuntiata, quae ad ipsam logicam pertinent, neque intra logicam constitui neque probari possunt. Ad talia probanda utimur probatione externa. Probationes externae interdum \u00abmetamathematicae\u00bb vocantur, et iam huiusmodi exemplum vidimus, cum (meta)theorema deductionis tractavimus. Hic est locus in quo probationes inductivae collocantur.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Probationes per Inductionem Mathematicam<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=eJQcNPrKyW0&amp;t=359s\" target=\"_blank\" rel=\"noopener\"><strong>Inductio Mathematica est methodus demonstrationis<\/strong><\/a> quae nobis permittit demonstrare quaedam valere pro omnibus numeris naturalibus.<\/p>\n<p style=\"text-align: justify;\"><strong>EXEMPLUM:<\/strong><br \/>\nProbare possumus omnem numerum formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^n - 4^n<\/span><\/span>, ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> est quivis numerus naturalis, semper divisibilem esse per 7.<br \/>\n<strong>DEMONSTRATIO:<\/strong> Si inspiciamus quid accidat cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>, videbimus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^1 - 4^1 = 7<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">quod, ut manifestum est, divisibile est per 7.<\/p>\n<p style=\"text-align: justify;\">Nunc supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^n - 4^n<\/span><\/span> divisibile esse pro quodam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k.<\/span><\/span> Ex hoc probabimus hanc expressionem etiam valere pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k+1.<\/span><\/span> Hoc fieri potest modo sequente:<\/p>\n<table\">\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^{k+1} - 4^{k+1}<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=11 \\cdot 11^{k} - 4 \\cdot 4^{k}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=11 \\cdot 11^{k} - (11-7) \\cdot 4^{k}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=11 \\cdot 11^{k} - 11 \\cdot 4^{k} + 7\\cdot 4^{k}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=11 ( 11^{k} - 4^{k} ) + 7\\cdot 4^{k}<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ergo, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^k - 4^k<\/span><\/span> divisibile est per 7, consequenter erit etiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11 ( 11^{k} - 4^{k} ) + 7\\cdot 4^{k}<\/span><\/span>, quod idem est ac dicere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^{k+1} - 4^{k+1}<\/span><\/span> divisibile esse per 7. Hinc habemus: si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11^k - 4^k<\/span><\/span> est divisibile pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=1<\/span><\/span>, tum etiam pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=2, k=3, k=4,\\cdots<\/span><\/span> et sic porro, atque proinde, divisibile pro quovis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{N}.<\/span><\/span> Cum hoc fit, dicimus inductionem esse completam. \u25a0<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Probationes per Inductionem in Logica Propositionali<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=eJQcNPrKyW0&amp;t=775s\" target=\"_blank\" rel=\"noopener\"><strong>Ad probationes per inductionem quas infra perficimus,<\/strong><\/a> necesse est prius introducere sequentem notationis conventionem.<\/p>\n<p style=\"text-align: justify;\"><strong>NOTATIO: <\/strong> Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_1,\\cdots, F_n<\/span><\/span> finitium expressiones quaelibet logicae propositionalis. Coniunctiones et disiunctiones harum expressionum introducuntur hoc modo:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigwedge_{i=1}^n F_i := F_1\\wedge \\cdots \\wedge F_n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigvee_{i=1}^n F_i := F_1\\vee \\cdots \\vee F_n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">His positis, nunc tractabimus duas formas generalizatas sequentis naturae.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Forma Generalizata Legum De Morgan<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=eJQcNPrKyW0&amp;t=829s\" target=\"_blank\" rel=\"noopener\"><strong>Dato quodam finito collectione expressionum logicae propositionalis<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_1,\\cdots, F_n,<\/span><\/span> semper valent sequentes duae proprietates:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\neg\\left(\\bigwedge_{i=1}^n F_i \\right) \\equiv \\left( \\bigvee_{i=1}^n \\neg F_i \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\neg\\left(\\bigvee_{i=1}^n F_i \\right) \\equiv \\left( \\bigwedge_{i=1}^n \\neg F_i \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><strong>DEMONSTRATIO:<\/strong> Primum demonstrabimus per inductionem super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\left(\\bigwedge_{i=1}^n F_i \\right) \\equiv \\left( \\bigvee_{i=1}^n \\neg F_i \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Primo inspiciendum est quid fiat in casu initiali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1.<\/span><\/span> Hoc in casu patet quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F_1 \\equiv \\neg\\left(\\bigwedge_{i=1}^1F_i\\right)\\equiv \\left(\\bigvee_{i=1}^n \\neg F_i \\right) \\equiv\\neg F_1<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc supponamus hanc proprietatem valere pro quodam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k;<\/span><\/span> id est, data quadam finita collectione expressionum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_1, F_2, \\cdots, F_k<\/span><\/span> valet quod:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\neg\\left(\\bigwedge_{i=1}^k F_i\\right) \\equiv \\left(\\bigvee_{i=1}^k \\neg F_i\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify\">Tunc probabimus hinc sequi<\/p>\n<p style=\"text-align: justify\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\neg\\left(\\bigwedge_{i=1}^{k+1} F_i\\right) \\equiv \\left(\\bigvee_{i=1}^{k+1} \\neg F_i\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify\">Definita conjunctione habemus:<\/p>\n<p style=\"text-align: justify\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\neg\\left(\\bigwedge_{i=1}^{k+1} F_i\\right) := \\neg\\left[\\left(\\bigwedge_{i=1}^{k} F_i\\right) \\wedge F_{k+1}\\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ad hanc expressionem applicare possumus leges De Morgan (illam usualem pro duobus terminis) ut consequamur:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\neg\\left(\\bigwedge_{i=1}^{k+1} F_i\\right)\\equiv \\left[\\neg\\left(\\bigwedge_{i=1}^{k} F_i\\right) \\vee \\neg F_{k+1}\\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc, si adhibeamus hypothesim inductionis, obtinebimus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\neg\\left(\\bigwedge_{i=1}^{k+1} F_i\\right)\\equiv \\left[ \\left(\\bigvee_{i=1}^k \\neg F_i\\right) \\vee \\neg F_{k+1}\\right] := \\left(\\bigvee_{i=1}^{k+1}\\neg F_i \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quare inductio est completa, et proprietas valet pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> in genere. Secunda relatio similiter demonstrari potest modo prorsus analogo, quam ob rem lectori relinquitur exercenda, muajaja!<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Forma Generalizata Legum Distributivarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=eJQcNPrKyW0&amp;t=1205s\" target=\"_blank\" rel=\"noopener\"><strong>Similiter ac in legibus De Morgan<\/strong><\/a>, leges distributionis generalizari possunt hoc modo. Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{F_1, \\cdots, F_n\\}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{G_1,\\cdots, G_m\\}<\/span><\/span> duae collectiones finitae expressionum quarumlibet; tunc valent hae aequivalentiae:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^m G_j \\right) \\right] \\equiv \\left[\\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^m(F_i\\vee G_j) \\right) \\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigvee_{i=1}^n F_i \\right) \\wedge \\left(\\bigvee_{j=1}^m G_j \\right) \\right] \\equiv \\left[\\bigvee_{i=1}^n\\left(\\bigvee_{j=1}^m(F_i\\wedge G_j) \\right) \\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><strong>DEMONSTRATIO:<\/strong> Ad hanc demonstrationem construendam oportet uti inductione duplici, super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m.<\/span><\/span> In sequentibus peragam inductionem primum super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>, deinde super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span>, pro expressione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^m G_j \\right) \\right] \\equiv \\left[\\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^m(F_i\\vee G_j) \\right) \\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Si ponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=1,<\/span><\/span> tunc haec expressio scribitur ut:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^1 G_j \\right) \\right] \\equiv \\left[\\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^1(F_i\\vee G_j) \\right) \\right].<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod idem est ac dicere:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee G_1 \\right] \\equiv \\left[\\bigwedge_{i=1}^n\\left( F_i\\vee G_1 \\right) \\right].<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc demonstrabimus hanc expressionem per inductionem super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Si ponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1,<\/span><\/span> tunc expressio ad haec reducitur:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_1 \\vee G_1 \\equiv F_1 \\vee G_1.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod plane verum est. Nunc autem supponamus hoc valere pro quodam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>; hoc est, hypothesi inductionis statuemus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^k F_i \\right) \\vee G_1 \\right] \\equiv \\left[\\bigwedge_{i=1}^k\\left( F_i\\vee G_1 \\right) \\right].<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ex hoc nunc probabimus id valere etiam pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k+1.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Definitio conjunctionis dat:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[\\left(\\bigwedge_{i=1}^{k+1}F_i \\right) \\vee G_1 \\right] := \\left[\\left(\\left(\\bigwedge_{i=1}^{k}F_i \\right)\\wedge F_{k+1} \\right) \\vee G_1 \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc, adhibita <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-distributione, habebimus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[\\left(\\bigwedge_{i=1}^{k+1}F_i \\right) \\vee G_1 \\right] \\equiv \\left[\\left(\\left(\\bigwedge_{i=1}^{k}F_i \\right)\\vee G_{1} \\right) \\wedge \\left(F_{k+1} \\vee G_1 \\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et hoc ipso loco uti possumus hypothesi inductionis ad obtinendum:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[\\left(\\bigwedge_{i=1}^{k+1}F_i \\right) \\vee G_1 \\right] \\equiv \\left[\\left(\\bigwedge_{i=1}^k\\left( F_i\\vee G_1 \\right) \\right) \\wedge \\left(F_{k+1} \\vee G_1 \\right) \\right] := \\left[\\bigwedge_{i=1}^{k+1}(F_{i}\\vee G_1) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Itaque, per inductionem probavimus quod pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{N}<\/span><\/span> valet:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right)\\vee G_1\\right] \\equiv \\left[\\bigwedge_{i=1}^n(F_i\\vee G_1)\\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Completa inductione super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>, comprobavimus valere casum initialem pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=1,<\/span><\/span> nunc solum restat completa inductio super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m.<\/span><\/span> Ad hoc faciendum statuimus hypothesim inductionis pro quodam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=l<\/span><\/span>, id est, valere sequentem aequivalentiam:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^l G_j \\right) \\right] \\equiv \\left[\\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^l(F_i\\vee G_j) \\right) \\right]<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et ex hoc demonstrabimus quod etiam valet pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=l+1.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ex definitione conjunctionis habemus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{l+1} G_j \\right) \\right] := \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\left(\\bigwedge_{j=1}^{l} G_j \\right) \\wedge G_{l+1}\\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Deinde, adhibita <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-distributione, consequitur:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{l+1} G_j \\right) \\right] \\equiv \\left[ \\left( \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left( \\bigwedge_{j=1}^l G_j \\right) \\right) \\wedge \\left( \\left( \\bigwedge_{i=1}^n F_i \\right)\\vee G_{l+1} \\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quare, utens hypothesi inductionis, scribere potes:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{l+1} G_j \\right) \\right] \\equiv \\left[ \\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^l(F_i\\vee G_j) \\right) \\wedge \\left( \\left( \\bigwedge_{i=1}^n F_i \\right)\\vee G_{l+1} \\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et nunc, si applicemus resultatum inductionis super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>, habebimus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{l+1} G_j \\right) \\right] \\equiv \\left[ \\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^l(F_i\\vee G_j) \\right) \\wedge \\left( \\bigwedge_{i=1}^n (F_i \\vee G_{l+1} )\\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod, tandem, ex definitione conjunctionis fit:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{l+1} G_j \\right) \\right] \\equiv \\left[ \\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^{l+1}(F_i\\vee G_j) \\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Itaque, inductio super <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span> est completa, et expressio<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[ \\left(\\bigwedge_{i=1}^n F_i \\right) \\vee \\left(\\bigwedge_{j=1}^{m} G_j \\right) \\right] \\equiv \\left[ \\bigwedge_{i=1}^n\\left(\\bigwedge_{j=1}^{m}(F_i\\vee G_j) \\right) \\right] <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">valet pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n,m\\in\\mathbb{N}<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Hoc iter per probationes per inductionem ostendit quomodo technicae rigidae demonstrationis mathematicae adhiberi possint non solum in ambitu numerorum naturalium, sed etiam in logica propositionali. Per inductionem, validitatem formarum generalizatarum legum De Morgan et legum distributivarum statuimus, ita corroborantes intellectum fundamentorum logicorum qui sub variis partibus scientiae mathematicae latent. Hic modus non solum est essentialis ad evolutionem facultatum cogitandi abstracte, sed etiam praebet instrumentum validum ad quaestiones complexas tractandas in mathematica et ultra eam.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Probationes per Inductionem: Regulae Generalizatae De Morgan et Distributio SUMMARIUMIn hac lectione agitur de probationibus per inductionem in mathematica et logica propositionali. Exponuntur duo genera probationum: probationes internae sive deductivae, quae in regulis logicae nituntur, et probationes externae sive metamathematicae, quae necessariae sunt ad enuntiata probanda quae ad ipsam logicam spectant. Inductio Mathematica introducitur ut [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28154,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33901","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-mathematica","category-logica-propositionalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Probationes per Inductionem: Generalizatio De Morgan et Distributio - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Omnia de modis probationum per inductionem in mathematica et logica perficiendis. 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