{"id":8019,"date":"2023-04-10T06:00:15","date_gmt":"2023-04-10T04:00:15","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8019"},"modified":"2023-03-22T19:56:13","modified_gmt":"2023-03-22T17:56:13","slug":"10-abr-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/10-abr-23\/","title":{"rendered":"Relaciones antisim\u00e9tricas"},"content":{"rendered":"<p>Usando el <a href=\"https:\/\/bit.ly\/3IVVqOT\">tipo de las relaciones binarias<\/a>, definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   antisimetrica :: Eq a => Rel a -> Bool\n<\/pre>\n<p>tal que <code>antisimetrica r<\/code> se verifica si la relaci\u00f3n <code>r<\/code> es antisim\u00e9trica; es decir, si (x,y) e (y,x) est\u00e1n relacionado, entonces x=y. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   antisimetrica (R ([1,2],[(1,2)]))        ==  True\n   antisimetrica (R ([1,2],[(1,2),(2,1)]))  ==  False\n   antisimetrica (R ([1,2],[(1,1),(2,1)]))  ==  True\n<\/pre>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nimport Relaciones_binarias (Rel(R))\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nantisimetrica :: Eq a => Rel a -> Bool\nantisimetrica (R (_,g)) =\n  null [(x,y) | (x,y) <- g, x \/= y, (y,x) `elem` g]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nantisimetrica2 :: Eq a => Rel a -> Bool\nantisimetrica2 (R (_,g)) =\n  and [(y,x) `notElem` g | (x,y) <- g, x \/= y]\n\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nantisimetrica3 :: Eq a => Rel a -> Bool\nantisimetrica3 (R (_,g)) =\n  all (\\(x, y) -> (y,x) `notElem` g || x == y) g\n\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nantisimetrica4 :: Eq a => Rel a -> Bool\nantisimetrica4 (R (u,g)) =\n  and [((x,y) `elem` g && (y,x) `elem` g) --> (x == y)\n       | x <- u, y <- u]\n  where p --> q = not p || q\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nantisimetrica5 :: Eq a => Rel a -> Bool\nantisimetrica5 (R (_,g)) = aux g\n  where aux []         = True\n        aux ((x,y):g') = ((y,x) `notElem` g || x == y) && aux g'\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_antisimetrica :: Rel Int -> Bool\nprop_antisimetrica r =\n  all (== antisimetrica r)\n      [antisimetrica2 r,\n       antisimetrica3 r,\n       antisimetrica4 r,\n       antisimetrica5 r]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_antisimetrica\n--    +++ OK, passed 100 tests.\n<\/pre>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom typing import TypeVar\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nfrom src.Relaciones_binarias import Rel, relacionArbitraria\n\nA = TypeVar('A')\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef antisimetrica(r: Rel[A]) -> bool:\n    (u, g) = r\n    return [(x, y) for (x, y) in g if x != y and (y, x) in g] == []\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef antisimetrica2(r: Rel[A]) -> bool:\n    (u, g) = r\n    return all(((y, x) not in g for (x, y) in g if x != y))\n\n# 3\u00aa soluci\u00f3n\n# ===========\n\ndef antisimetrica3(r: Rel[A]) -> bool:\n    (u, g) = r\n    return all ((not ((x, y) in g and (y, x) in g) or x == y\n                 for x in u for y in u))\n\n# 4\u00aa soluci\u00f3n\n# ===========\n\ndef antisimetrica4(r: Rel[A]) -> bool:\n    (u, g) = r\n    def aux(xys: list[tuple[A, A]]) -> bool:\n        if not xys:\n            return True\n        (x, y) = xys[0]\n        return ((y, x) not in g or x == y) and aux(xys[1:])\n\n    return aux(g)\n\n# 5\u00aa soluci\u00f3n\n# ===========\n\ndef antisimetrica5(r: Rel[A]) -> bool:\n    (u, g) = r\n    for (x, y) in g:\n        if (y, x) in g and x != y:\n            return False\n    return True\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=0, max_value=10))\ndef test_antisimetrica(n: int) -> None:\n    r = relacionArbitraria(n)\n    res = antisimetrica(r)\n    assert antisimetrica2(r) == res\n    assert antisimetrica3(r) == res\n    assert antisimetrica4(r) == res\n    assert antisimetrica5(r) == res\n\n# La comprobaci\u00f3n es\n#    > poetry run pytest -q Relaciones_antisimetricas.py\n#    1 passed in 0.13s\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Usando el tipo de las relaciones binarias, definir la funci\u00f3n antisimetrica :: Eq a => Rel a -> Bool tal que antisimetrica r se verifica si la relaci\u00f3n r es antisim\u00e9trica; es decir, si (x,y) e (y,x) est\u00e1n relacionado, entonces x=y. Por ejemplo, antisimetrica (R ([1,2],[(1,2)])) == True antisimetrica (R ([1,2],[(1,2),(2,1)])) == False antisimetrica (R&#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":[581],"tags":[576],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8019"}],"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=8019"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8019\/revisions"}],"predecessor-version":[{"id":8020,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8019\/revisions\/8020"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}