{"id":5344,"date":"2020-01-10T05:30:16","date_gmt":"2020-01-10T03:30:16","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5344"},"modified":"2022-06-06T12:09:18","modified_gmt":"2022-06-06T10:09:18","slug":"teorema-de-existencia-de-divisores","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/teorema-de-existencia-de-divisores\/","title":{"rendered":"Teorema de existencia de divisores"},"content":{"rendered":"<p>El <strong>teorema de existencia de divisores<\/strong> afirma que<\/p>\n<blockquote><p>\nEn cualquier subconjunto de {1, 2, &#8230;, 2m} con al menos m+1 elementos existen n\u00fameros distintos a, b tales que a divide a b.\n<\/p><\/blockquote>\n<p>Un conjunto de n\u00fameros naturales xs es mayoritario si existe un m tal que la lista de xs es un subconjunto de {1,2,&#8230;,2m} con al menos m+1 elementos. Por ejemplo, {2,3,5,6} porque es un subconjunto de {1,2,&#8230;,6} con m\u00e1s de 3 elementos.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   divisoresMultiplos :: [Integer] -> [(Integer,Integer)]\n   esMayoritario :: [Integer] -> Bool\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(divisores xs) es la lista de pares de elementos distintos de (a,b) tales que a divide a b. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">  \n     divisoresMultiplos [2,3,5,6]  ==  [(2,6),(3,6)]\n     divisoresMultiplos [2,3,5]    ==  []\n     divisoresMultiplos [4..8]     ==  [(4,8)]\n<\/pre>\n<ul>\n<li>(esMayoritario xs) se verifica xs es mayoritario. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     esMayoritario [2,3,5,6]  ==  True\n     esMayoritario [2,3,5]    ==  False\n<\/pre>\n<p>Comprobar con QuickCheck el teorema de existencia de divisores; es decir, en cualquier conjunto mayoritario existen n\u00fameros distintos a, b tales que a divide a b. Para la comprobaci\u00f3n se puede usar el siguiente generador de conjuntos mayoritarios<\/p>\n<pre lang=\"text\">\n   mayoritario :: Gen [Integer]\n   mayoritario = do\n     m' <- arbitrary\n     let m = 1 + abs m'\n     xs' <- sublistOf [1..2*m] `suchThat` (\\ys -> genericLength ys > m)\n     return xs'\n<\/pre>\n<p>con lo que la propiedad que hay que comprobar con QuickCheck es<\/p>\n<pre lang=\"text\">\n   teorema_de_existencia_de_divisores :: Property\n   teorema_de_existencia_de_divisores =\n     forAll mayoritario (not . null . divisoresMultiplos)\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\nimport Test.QuickCheck\n\ndivisoresMultiplos :: [Integer] -> [(Integer,Integer)]\ndivisoresMultiplos xs =\n  [(x,y) | x <- xs\n         , y <- xs\n         , y \/= x\n         , y `mod` x == 0]\n\nesMayoritario :: [Integer] -> Bool\nesMayoritario xs =\n  not (null xs) && length xs > ceiling (n \/ 2) \n  where n = fromIntegral (maximum xs)\n\n-- Comprobaci\u00f3n del teorema\n-- ========================\n\n-- La propiedad es\nteorema_de_existencia_de_divisores :: Property\nteorema_de_existencia_de_divisores =\n  forAll mayoritario (not . null . divisoresMultiplos)\n\n-- mayoritario es un generador de conjuntos mayoritarios. Por ejemplo, \n--    \u03bb> sample mayoritario\n--    [1,2]\n--    [2,5,7,8]\n--    [1,2,8,10,14]\n--    [3,8,11,12,13,15,18,19,22,23,25,26]\n--    [1,3,4,6]\n--    [3,6,9,11,12,14,17,19]\nmayoritario :: Gen [Integer]\nmayoritario = do\n  m' <- arbitrary\n  let m = 1 + abs m'\n  xs' <- sublistOf [1..2*m] `suchThat` (\\ys -> genericLength ys > m)\n  return xs'\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck teorema_de_existencia_de_divisores\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nGuiomar, Guiomar,<br \/>\nm\u00edrame en ti castigado:<br \/>\nreo de haberte creado,<br \/>\nya no te puedo olvidar.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El teorema de existencia de divisores afirma que En cualquier subconjunto de {1, 2, &#8230;, 2m} con al menos m+1 elementos existen n\u00fameros distintos a, b tales que a divide a b. Un conjunto de n\u00fameros naturales xs es mayoritario si existe un m tal que la lista de xs es un subconjunto de {1,2,&#8230;,2m}&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[7],"tags":[130,483,322,8,486,338,183,28,15,89,181,141,371,485,539,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5344"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=5344"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5344\/revisions"}],"predecessor-version":[{"id":5406,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5344\/revisions\/5406"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5344"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5344"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}