{"id":7493,"date":"2022-11-03T06:00:20","date_gmt":"2022-11-03T04:00:20","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7493"},"modified":"2022-12-14T12:16:23","modified_gmt":"2022-12-14T10:16:23","slug":"subconjuntos-de-un-conjunto","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/subconjuntos-de-un-conjunto\/","title":{"rendered":"Subconjuntos de un conjunto"},"content":{"rendered":"

Definir la funci\u00f3n<\/p>\n

\n   subconjuntos :: [a] -> [[a]]\n<\/pre>\n
tal que subconjuntos xs<\/code> es la lista de las subconjuntos de la lista xs<\/code>. Por ejemplo,<\/p>\n
\n   \u03bb> subconjuntos [2,3,4]\n   [[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4],[]]\n   \u03bb> subconjuntos [1,2,3,4]\n   [[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],\n      [2,3,4],  [2,3],  [2,4],  [2],  [3,4],  [3],  [4], []]\n<\/pre>\n
Comprobar con QuickChek que el n\u00famero de elementos de subconjuntos xs<\/code> es 2 elevado al n\u00famero de elementos de xs<\/code>.<\/p>\n
Soluciones<\/b><\/p>\n
A continuaci\u00f3n se muestran las soluciones en Haskell<\/a> y las soluciones en Python<\/a>.<\/p>\n
<\/a>
\nSoluciones en Haskell<\/b><\/p>\n
\nimport Data.List (sort, subsequences)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntos1 :: [a] -> [[a]]\nsubconjuntos1 []     = [[]]\nsubconjuntos1 (x:xs) = [x:ys | ys <- sub] ++ sub\n  where sub = subconjuntos1 xs\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntos2 :: [a] -> [[a]]\nsubconjuntos2 []     = [[]]\nsubconjuntos2 (x:xs) = map (x:) sub ++ sub\n  where sub = subconjuntos2 xs\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntos3 :: [a] -> [[a]]\nsubconjuntos3 = subsequences\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_subconjuntos :: [Int] -> Bool\nprop_subconjuntos xs =\n  all (== sort (subconjuntos1 xs))\n      [sort (subconjuntos2 xs),\n       sort (subconjuntos3 xs)]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheckWith (stdArgs {maxSize=7}) prop_subconjuntos\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (subconjuntos1 [1..23])\n--    8388608\n--    (2.05 secs, 1,476,991,840 bytes)\n--    \u03bb> length (subconjuntos2 [1..23])\n--    8388608\n--    (0.87 secs, 1,208,555,312 bytes)\n--    \u03bb> length (subconjuntos3 [1..23])\n--    8388608\n--    (0.09 secs, 873,006,608 bytes)\n\n-- Comprobaci\u00f3n de la propiedad\n-- ============================\n\n-- La propiedad es\nprop_length_subconjuntos :: [Int] -> Bool\nprop_length_subconjuntos xs =\n  length (subconjuntos1 xs) == 2 ^ length xs\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheckWith (stdArgs {maxSize=7}) prop_length_subconjuntos\n--    +++ OK, passed 100 tests.\n<\/pre>\n
<\/a>
\nSoluciones en Python<\/b><\/p>\n
\nfrom itertools import combinations\nfrom sys import setrecursionlimit\nfrom timeit import Timer, default_timer\nfrom typing import TypeVar\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\nfrom sympy import FiniteSet\n\nsetrecursionlimit(10**6)\n\nA = TypeVar('A')\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef subconjuntos1(xs: list[A]) -> list[list[A]]:\n    if xs:\n        sub = subconjuntos1(xs[1:])\n        return [[xs[0]] + ys for ys in sub] + sub\n    return [[]]\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef subconjuntos2(xs: list[A]) -> list[list[A]]:\n    if xs:\n        sub = subconjuntos1(xs[1:])\n        return list(map((lambda ys: [xs[0]] + ys), sub)) + sub\n    return [[]]\n\n# 3\u00aa soluci\u00f3n\n# ===========\n\ndef subconjuntos3(xs: list[A]) -> list[list[A]]:\n    c = FiniteSet(*xs)\n    return list(map(list, c.powerset()))\n\n# 4\u00aa soluci\u00f3n\n# ===========\n\ndef subconjuntos4(xs: list[A]) -> list[list[A]]:\n    return [list(ys)\n            for r in range(len(xs)+1)\n            for ys in combinations(xs, r)]\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.lists(st.integers(), max_size=5))\ndef test_subconjuntos(xs: list[int]) -> None:\n    ys = list(set(xs))\n    r = sorted([sorted(zs) for zs in subconjuntos1(ys)])\n    assert sorted([sorted(zs) for zs in subconjuntos2(ys)]) == r\n    assert sorted([sorted(zs) for zs in subconjuntos3(ys)]) == r\n    assert sorted([sorted(zs) for zs in subconjuntos4(ys)]) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q subconjuntos_de_un_conjunto.py\n#    1 passed in 0.89s\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e: str) -> None:\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('subconjuntos1(range(14))')\n#    0.00 segundos\n#    >>> tiempo('subconjuntos2(range(14))')\n#    0.00 segundos\n#    >>> tiempo('subconjuntos3(range(14))')\n#    6.01 segundos\n#    >>> tiempo('subconjuntos4(range(14))')\n#    0.00 segundos\n#\n#    >>> tiempo('subconjuntos1(range(23))')\n#    1.95 segundos\n#    >>> tiempo('subconjuntos2(range(23))')\n#    2.27 segundos\n#    >>> tiempo('subconjuntos4(range(23))')\n#    1.62 segundos\n\n# Comprobaci\u00f3n de la propiedad\n# ============================\n\n# La propiedad es\n@given(st.lists(st.integers(), max_size=7))\ndef test_length_subconjuntos(xs: list[int]) -> None:\n    assert len(subconjuntos1(xs)) == 2 ** len(xs)\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q subconjuntos_de_un_conjunto.py\n#    2 passed in 0.95s\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Definir la funci\u00f3n subconjuntos :: [a] -> [[a]] tal que subconjuntos xs es la lista de las subconjuntos de la lista xs. Por ejemplo, \u03bb> subconjuntos [2,3,4] [[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4],[]] \u03bb> subconjuntos [1,2,3,4] [[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1], [2,3,4], [2,3], [2,4], [2], [3,4], [3], [4], []] Comprobar con QuickChek que el n\u00famero de elementos de subconjuntos xs es 2 elevado al…<\/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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7493"}],"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=7493"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7493\/revisions"}],"predecessor-version":[{"id":7656,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7493\/revisions\/7656"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7493"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}