{"id":2691,"date":"2020-09-20T10:50:01","date_gmt":"2020-09-20T08:50:01","guid":{"rendered":"https:\/\/blog.atlant.is\/?p=2691"},"modified":"2021-02-21T13:31:57","modified_gmt":"2021-02-21T11:31:57","slug":"0-1","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=2691","title":{"rendered":"0! = 1"},"content":{"rendered":"\n<p>La fonction factorielle est souvent d\u00e9finie comme le produit&nbsp;: $$n!=\\prod_{k=1}^n k\\text{,}$$ ou bien par r\u00e9currence&nbsp;: $$n!=n\\cdot(n-1)!\\text{,}$$ en prenant par convention&nbsp;: $0!=1$. Mais pourquoi cette convention&nbsp;?<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/zero.png\" alt=\"\" class=\"wp-image-3445\" width=\"375\" height=\"375\" srcset=\"https:\/\/blog.douzeb.is\/wp-content\/uploads\/2020\/09\/zero.png 500w, https:\/\/blog.douzeb.is\/wp-content\/uploads\/2020\/09\/zero-300x300.png 300w, https:\/\/blog.douzeb.is\/wp-content\/uploads\/2020\/09\/zero-150x150.png 150w\" sizes=\"(max-width: 375px) 100vw, 375px\" \/><figcaption>Zero<\/figcaption><\/figure><\/div>\n\n\n\n<!--more-->\n\n\n\n<p>Bien s\u00fbr c&rsquo;est assez pratique pour la formule des <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Combinaison_(math%C3%A9matiques)\">combinaisons<\/a>, car ainsi on peut \u00e9crire&nbsp;: $$C_n^k=\\left(\\begin{array}{c}n\\\\k\\end{array}\\right)={n!\\over k!(n-k)!}$$ m\u00eame quand $k=n$, et v\u00e9rifier qu&rsquo;il y a bien une seule fa\u00e7on de choisir $n$ \u00e9l\u00e9ments parmi $n$ \ud83d\ude42<\/p>\n\n\n\n<p>On peut trouver une explication plus satisfaisante en revenant au sens ensembliste de la factorielle&nbsp;: $n!$ c&rsquo;est le nombre de permutations d&rsquo;un ensemble \u00e0 $n$ \u00e9l\u00e9ments, c&rsquo;est \u00e0 dire le nombre de bijections possibles entre cet ensemble et lui-m\u00eame.<\/p>\n\n\n\n<p>Du coup la question devient : quelles sont les bijections entre l&rsquo;ensemble vide $\\emptyset$ et lui-m\u00eame ? Quel est leur nombre ?<\/p>\n\n\n\n<p>Pour revenir aux bases, une application $f$ d&rsquo;un ensemble $A$ vers un ensemble $B$ est une relation entre $A$ et $B$ &#8212; autrement dit une partie $f\\subset A\\times B$ &#8212; qui v\u00e9rifie&nbsp;: $$\\forall x\\in A, \\exists! y\\in B, (x,y)\\in f\\text{.}$$ Et dans le cas o\u00f9 $(x,y)\\in f$ on \u00e9crit&nbsp;: $y=f(x)$.<\/p>\n\n\n\n<p>On observe que $\\emptyset\\times\\emptyset=\\emptyset$, que donc $\\emptyset$ est la seule partie de $\\emptyset\\times\\emptyset$, et on v\u00e9rifie que $\\forall x\\in \\emptyset, \\exists! y\\in \\emptyset, (x,y)\\in \\emptyset$, ce qui d\u00e9montre que $\\emptyset$ est la seule application de $\\emptyset$ vers $\\emptyset$. On remarque que le domaine de (l&rsquo;application) $\\emptyset$ est vide, et que donc on n&rsquo;a jamais l&rsquo;occasion d&rsquo;\u00e9crire $\\emptyset(x)=y$.<\/p>\n\n\n\n<p>Pour finir, on constate que (l&rsquo;application) $\\emptyset:\\emptyset\\to\\emptyset$ est \u00e0 la fois injective (son domaine est vide) et surjective (son ensemble d&rsquo;arriv\u00e9e est vide), ce qui d\u00e9montre que $\\emptyset$ est une bijection.<\/p>\n\n\n\n<p>Il est temps de conclure&nbsp;: $\\emptyset$ est l&rsquo;unique bijection entre $\\emptyset$ et $\\emptyset$&#8230;<\/p>\n\n\n\n<p>$0!=1$.$\\quad \\Box$ <\/p>\n","protected":false},"excerpt":{"rendered":"<p>La fonction factorielle est souvent d\u00e9finie comme le produit&nbsp;: $$n!=\\prod_{k=1}^n k\\text{,}$$ ou bien par r\u00e9currence&nbsp;: $$n!=n\\cdot(n-1)!\\text{,}$$ en prenant par convention&nbsp;: $0!=1$. Mais pourquoi cette convention&nbsp;?<\/p>\n","protected":false},"author":1,"featured_media":3445,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[16,19],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2691"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2691"}],"version-history":[{"count":24,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2691\/revisions"}],"predecessor-version":[{"id":3490,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2691\/revisions\/3490"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/3445"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2691"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2691"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2691"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=2691"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}