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"fromtitle": "Accueil",
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"totitle": "Alg\u00e8bre/Th\u00e9orie \u00e9l\u00e9mentaire des ensembles",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Ligne\u00a01\u202f:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Ligne\u00a01\u202f:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"><strong>MediaWiki a \u00e9t\u00e9 install\u00e9.</strong></del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Ensembles ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== D\u00e9finitions : ensemble et \u00e9l\u00e9ment ===</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Consultez le [https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special:MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Help:Contents Guide </del>de <del class=\"diffchange diffchange-inline\">l\u2019utilisateur] pour plus d\u2019informations sur l\u2019utilisation </del>de ce <del class=\"diffchange diffchange-inline\">logiciel de wiki</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Les objets math\u00e9matiques sont nomm\u00e9s '''\u00e9l\u00e9ments'''</ins>. <ins class=\"diffchange diffchange-inline\">Un '''ensemble''' est une ''collection'' ou un ''groupement'' d'\u00e9l\u00e9ments</ins>. <ins class=\"diffchange diffchange-inline\"><br </ins>/<ins class=\"diffchange diffchange-inline\">>Soit <math>E<</ins>/<ins class=\"diffchange diffchange-inline\">math> un ensemble, quand <math>a<</ins>/<ins class=\"diffchange diffchange-inline\">math> est un \u00e9l\u00e9ment </ins>de <ins class=\"diffchange diffchange-inline\"><math>E</math>, nous disons que <math>a</math> est dans <math>E</math> ou que <math>a</math> appartient \u00e0 <math>E</math> et nous \u00e9crivons <math>a \\in E</math>, ce qui se lit \u00ab&nbsp;<math>a</math> appartient \u00e0 <math>E</math>&nbsp;\u00bb. Quant au contraire <math>a</math> n'est pas \u00e9l\u00e9ment </ins>de <ins class=\"diffchange diffchange-inline\"><math>E</math>, nous disons que <math>a</math> n'appartient pas \u00e0 <math>E</math> et nous \u00e9crivons <math>a \\not\\in E</math>, </ins>ce <ins class=\"diffchange diffchange-inline\">qui se lit \u00ab&nbsp;<math>a</math> n'appartient pas \u00e0 <math>E</math>&nbsp;\u00bb</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Pour d\u00e9marrer </del>==</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>==<ins class=\"diffchange diffchange-inline\">= D\u00e9finition/Notation : ensemble vide =</ins>==</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Configuration_settings Liste </del>des <del class=\"diffchange diffchange-inline\">param\u00e8tres </del>de <del class=\"diffchange diffchange-inline\">configuration]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">FAQ Questions courantes sur MediaWiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Un ensemble est dit vide s'il n'a aucun \u00e9l\u00e9ment et nous notons l<nowiki>'</nowiki>'''ensemble vide''' <math>\\varnothing</math>. </ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">postorius</del>/<del class=\"diffchange diffchange-inline\">lists</del>/<del class=\"diffchange diffchange-inline\">mediawiki-announce</del>.<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/ <del class=\"diffchange diffchange-inline\">Liste </del>de <del class=\"diffchange diffchange-inline\">discussion sur les distributions </del>de <del class=\"diffchange diffchange-inline\">MediaWiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Adaptez MediaWiki dans votre langue</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:'''Remarque''' </ins>: <ins class=\"diffchange diffchange-inline\">retenons qu'une chose est un ensemble, si nous pouvons dire si un objet quelconque est ou n'est pas \u00e9l\u00e9ment de cette chose; concernant l'ensemble vide nous pouvons dire qu' aucun objet n'est \u00e9l\u00e9ment de cette chose.</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Combating_spam Apprendre comment combattre le pourriel dans votre wiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Exemples d'ensembles ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#Les entiers naturels <math>0, 1, 2, 3, ...<</ins>/<ins class=\"diffchange diffchange-inline\">math> forment un ensemble qui se note <math>\\mathbb{N}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#Les entiers relatifs <math>..., -3, -2, -1, 0, 1, 2, 3, .</ins>..<ins class=\"diffchange diffchange-inline\"><</ins>/<ins class=\"diffchange diffchange-inline\">math> forment un ensemble qui se note <math>\\mathbb{Z}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#Les nombres rationnels (de la forme <math>p/q</math> o\u00f9 <math>p \\in \\mathbb{Z}</math> et <math>q \\in \\mathbb{N}^*</math>) forment un ensemble not\u00e9 <math>\\mathbb{Q}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#Les points du plan forment un ensemble.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== D\u00e9finition d'un ensemble en extension et en compr\u00e9hension ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Un ensemble peut \u00eatre d\u00e9fini '''en extension''', c'est-\u00e0-dire en donnant la liste de ses \u00e9l\u00e9ments entre accolades, ou '''en compr\u00e9hension''' c'est-\u00e0-dire par une propri\u00e9t\u00e9 caract\u00e9risant ses \u00e9l\u00e9ments.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">La mani\u00e8re la plus simple de d\u00e9crire un ensemble \u00ab fini \u00bb est de lister ses \u00e9l\u00e9ments entre accolades. L'ensemble est alors d\u00e9fini en extension. Par exemple {1,2} repr\u00e9sente l'ensemble dont les \u00e9l\u00e9ments sont 1 et 2. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*L' ordre des \u00e9l\u00e9ments ne rev\u00eat aucune importance; par exemple, {1,2} = {2,1}. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*La r\u00e9p\u00e9tition d' \u00e9l\u00e9ments entre les accolades ne modifie pas l'ensemble; par exemple, {1,2,2} = {1,1,1,2} = {1,2}. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Pour d\u00e9finir en extension un ensemble dont le \u00ab nombre \u00bb d'\u00e9l\u00e9ments est \u00ab infini \u00bb, nous pouvons \u00e9crire quelques \u00e9l\u00e9ments de cet ensemble suivis de points de suspension. Par exemple, l'ensemble des entiers naturels se d\u00e9finit par </ins>: <ins class=\"diffchange diffchange-inline\"><math>\\mathbb{N}<</ins>/<ins class=\"diffchange diffchange-inline\">math>={0, 1, 2, 3, ...}. Les points de suspension peuvent aussi \u00eatre utilis\u00e9s pour abr\u00e9ger l'\u00e9criture de la liste des \u00e9l\u00e9ments de certains ensembles \u00ab finis \u00bb. Par exemple l'ensemble {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21} s'\u00e9crit plus simplement {1, 3, 5,..., 21}. Un abus de notation permet de d\u00e9finir un ensemble en pla\u00e7ant entre accolades la nature des objets qui lui appartiennent. Par exemple la notation {entiers pairs} d\u00e9signe l'ensemble de tous les entiers relatifs multiples de 2.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Il est aussi possible de d\u00e9finir un ensemble par une proposition logique ''P'' qui d\u00e9pend de ''x''. L'ensemble est alors constitu\u00e9 de tous les objets ''x'' pour lesquels la condition ''P'' est vraie. Cet ensemble se note {''x'' / ''P''(''x'')}. Par exemple, {''x''/x est un nombre r\u00e9el} d\u00e9signe l'ensemble des nombre r\u00e9els <math>\\mathbb{R}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Cette notation est appel\u00e9e \u00ab notation de d\u00e9finition d'un ensemble en compr\u00e9hension \u00bb. Quelques variantes de notations de d\u00e9finition d'un ensemble en compr\u00e9hension sont </ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{''x'' \u2208 ''A'' / ''P''(''x'')} d\u00e9signe l'ensemble des ''x'' qui sont d\u00e9j\u00e0 \u00e9l\u00e9ments de ''A'' tels que la condition ''P'' soit v\u00e9rifi\u00e9e pour ces ''x''. Par exemple, si <math>\\mathbb{N}</math> est l'ensemble </ins>des <ins class=\"diffchange diffchange-inline\">entiers, alors {''x'' \u2208 <math>\\mathbb{N}</math> / ''x'' est un entier pair} est l'ensemble </ins>de <ins class=\"diffchange diffchange-inline\">tous les entiers pairs.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{''F''(''x'') / ''x'' \u2208 ''A''} d\u00e9signe l'ensemble de tous les objets obtenus en mettant les \u00e9l\u00e9ments de l'ensemble ''A'' dans la formule ''F''. Par exemple, {2''x'' / ''x'' \u2208 <math>\\mathbb{N}</math>} est encore l'ensemble de tous les entiers pairs.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">{''F''(''x'') / ''P''(''x'')} est la forme la plus g\u00e9n\u00e9rale de la d\u00e9finition en compr\u00e9hension. Par exemple, { propri\u00e9taire de x / x est un chien} est l'ensemble de tous les propri\u00e9taires de chiens.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== D\u00e9finition : \u00c9galit\u00e9 de deux ensembles ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Deux ensembles <math>E</math> et <math>F</math> sont dits \u00e9gaux s'ils ont exactement les m\u00eames \u00e9l\u00e9ments et nous \u00e9crivons <math>E = F</math>. Nous avons</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\forall x, (x\\in E \\Leftrightarrow x\\in F)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>[<ins class=\"diffchange diffchange-inline\">[Alg\u00e8bre|retour \u00e0 l'alg\u00e8bre]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Sous-ensemble, partie d'un ensemble ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Inclusion ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soient <math>E</math> et <math>F</math> deux ensembles quelconques. Nous disons que <math>E</math> est <u>inclus</u> dans <math>F</math> ou que <math>E</math> est un <u>sous-ensemble</u> de <math>F</math> ou encore que <math>E</math> est une <u>partie</u> de <math>F</math> si tout \u00e9l\u00e9ment de <math>E</math> est un \u00e9l\u00e9ment de <math>F</math>. Nous \u00e9crivons <math>E\\subset F</math>.<br />Soit </ins>: <ins class=\"diffchange diffchange-inline\"><math>(E \\subset F) \\Leftrightarrow (\\forall x \\in E, x \\in F)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:''Exemple :'' <math>\\mathbb R\\subset \\mathbb C</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Notation'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nous notons <math>\\mathcal P(E)<</ins>/<ins class=\"diffchange diffchange-inline\">math>, l'ensemble des parties de l'ensemble <math>E</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Propositions'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#<math>(\\forall E, \\forall F, \\left((E\\subset F) \\land (F\\subset G)\\right))\\Rightarrow (E\\subset G)</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">#<math>(\\forall E, \\forall F, \\left((E\\subset F) \\land (F\\subset E\\right)))\\Leftrightarrow (E=F)<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::''D\u00e9monstrations :''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::1</ins>. <ins class=\"diffchange diffchange-inline\">Soient <math>E<</ins>/<ins class=\"diffchange diffchange-inline\">math>,<math>F<</ins>/<ins class=\"diffchange diffchange-inline\">math> et <math>G</math> trois ensembles.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:</ins>:<ins class=\"diffchange diffchange-inline\">Supposons <math>E \\subset F<</ins>/<ins class=\"diffchange diffchange-inline\">math> et <math>F \\subset G</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">:Soit <math>x \\in E</math>, on a <math>x \\in F </math> (car <math>E \\subset F</math>)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::De m\u00eame comme <math>x \\in F</math> et <math>F \\subset G</math> on a <math>x \\in G</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::Donc si <math>x \\in E</math> alors <math>x \\in G</math> d'o\u00f9 <math>E \\subset G</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:: 2. Soient <math>E</math> et <math>F</math> deux ensembles. L'\u00e9nonc\u00e9 <math>\\left((E\\subset F) \\land (F\\subset E\\right))</math> \u00e9quivaut \u00e0 dire que pour tout <math>x</math> on a <math>x\\in E \\Leftrightarrow x\\in F</math>. Finalement la propri\u00e9t\u00e9 annonc\u00e9e est une reformulation de l'[[Fondements des math\u00e9matiques/Les axiomes des th\u00e9ories des ensembles|axiome d'extensionalit\u00e9]</ins>]<ins class=\"diffchange diffchange-inline\">.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[</ins>[<ins class=\"diffchange diffchange-inline\">Alg\u00e8bre|retour \u00e0 l'alg\u00e8bre]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Op\u00e9rations sur les ensembles ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Intersection ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition </ins>:<ins class=\"diffchange diffchange-inline\">'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nous appelons '''intersection''' de deux ensembles quelconques ''E'' et ''F'', l'ensemble des ''x'' qui appartiennent \u00e0 la fois \u00e0 ''E'' et ''F''. Cet ensemble se note <math>E\\cap F<</ins>/<ins class=\"diffchange diffchange-inline\">math>, et nous avons</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>E\\cap F=\\{x/(x\\in E)\\land (x\\in F)\\}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>E\\cap F<</ins>/<ins class=\"diffchange diffchange-inline\">math> se lit \u00ab&nbsp;''E'' inter ''F''&nbsp;\u00bb.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Exemple :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Si ''A''={2,3,5,9} et ''B''={0,2,3}, alors leur intersection, est l'ensemble {2,3}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soient ''E'' et ''F'' deux ensembles quelconques. ''E'' et ''F'' sont dits '''disjoints''', lorsque leur intersection est vide, c'est-\u00e0-dire</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>E\\cap F=\\emptyset</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''Remarque :''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Il ne faut surtout pas confondre ''distincts'' avec ''disjoints''. Deux ensembles disjoints n'ont pas d'\u00e9l\u00e9ment en commun, alors que deux ensembles distincts peuvent en avoir. Pour que deux ensembles soient distincts il faut et il suffit qu'il existe un \u00e9l\u00e9ment appartenant \u00e0 l'un mais pas \u00e0 l'autre.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== R\u00e9union ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nous appelons '''r\u00e9union''' de deux ensembles ''E'' et ''F'' l'ensemble des ''x'' qui appartiennent \u00e0 ''E'' ou \u00e0 ''F'' (\u00e9ventuellement les deux). Cet ensemble se note <math>E\\cup F</math> et nous avons</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>E\\cup F=\\{x/(x\\in E)\\lor (x\\in F)\\}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>E\\cup F</math> se lit \u00ab&nbsp;''E'' union ''F''&nbsp;\u00bb.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Exemple :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Si ''A''={2,3,5,7} et ''B''={0,2,3}, alors leur r\u00e9union est l'ensemble {0,2,3,5,7}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Diff\u00e9rence ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soient ''E'' et ''F'' deux ensembles quelconques</ins>. <ins class=\"diffchange diffchange-inline\">Nous appelons '''diff\u00e9rence''' de ''E'' et ''F'', l'ensemble des ''x'' qui appartiennent \u00e0 ''E'' mais pas \u00e0 ''F''</ins>. <ins class=\"diffchange diffchange-inline\">Cet ensemble se note <math>E \\backslash F<</ins>/<ins class=\"diffchange diffchange-inline\">math> et nous avons </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>E\\backslash F=\\{x</ins>/<ins class=\"diffchange diffchange-inline\">(x\\in E) \\land (x\\notin F)\\}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>E\\backslash F</math> se lit \u00ab&nbsp;''E'' diff\u00e9rence ''F''&nbsp;\u00bb.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Diff\u00e9rence sym\u00e9trique ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soient ''E'' et ''F'' deux ensembles quelconques</ins>. <ins class=\"diffchange diffchange-inline\">Nous appelons '''diff\u00e9rence sym\u00e9trique''' de ''E'' et ''F'', l'ensemble des ''x'' qui appartiennent \u00e0 ''E'' ou \u00e0 ''F'' mais pas au deux \u00e0 la fois</ins>. <ins class=\"diffchange diffchange-inline\">Cet ensemble se note <math>E \\Delta F</math> et nous avons</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>E\\Delta F=\\{x/\\left((x\\in E)\\land (x\\notin F)\\right)\\lor \\left((x\\in F)\\land (x\\notin E)\\right) \\}=(E\\cup F)\\backslash (E\\cap F)</math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>E\\Delta F<</ins>/<ins class=\"diffchange diffchange-inline\">math> se lit \u00ab&nbsp;''E'' delta ''F''&nbsp;\u00bb.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Compl\u00e9mentaire ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finition :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soient ''E'' un ensemble quelconque et ''A'' une partie quelconque de ''E''. Nous appelons compl\u00e9mentaire de ''A'' par rapport \u00e0 ''E'' (ou de ''A'' dans ''E'') ou encore diff\u00e9rence </ins>de <ins class=\"diffchange diffchange-inline\">''E'' et </ins>de <ins class=\"diffchange diffchange-inline\">''A'', l'ensemble des ''x'' qui appartiennent \u00e0 ''E'' mais pas \u00e0 ''A''. Cet ensemble se note <math>\\complement_E A</math> ou <math>E \\backslash A</math> ou <math>\\overline{A}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Alg\u00e8bre|retour \u00e0 l'alg\u00e8bre]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Propri\u00e9t\u00e9s des op\u00e9rations \u00e9l\u00e9mentaires ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Propositions :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*L'intersection et la r\u00e9union sont idempotentes :<br/><math>\\forall A, A\\cap A=A</math><br/><math>\\forall A, A\\cup A=A</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*L'intersection et la r\u00e9union sont commutatives :<br/><math>\\forall A, \\forall B, A\\cap B=B\\cap A</math><br/><math>\\forall A, \\forall B, A\\cup B=B\\cup A</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*L'intersection et la r\u00e9union sont associatives :<br/><math>\\forall A, \\forall B, \\forall C, (A\\cap B)\\cap C=A\\cap (B\\cap C)</math><br/><math>\\forall A, \\forall B, \\forall C, (A\\cup B)\\cup C=A\\cup (B\\cup C)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*L'intersection est distributive par rapport \u00e0 la r\u00e9union :<br/><math>\\forall A, \\forall B, \\forall C, A\\cap (B\\cup C)=(A\\cap B)\\cup (A\\cap C)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*La r\u00e9union est distributive par rapport \u00e0 l'intersection : <br/><math>\\forall A, </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\forall B, \\forall C, A\\cup (B\\cap C)=(A\\cup B)\\cap (A\\cup C)</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Demonstrations :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soit A un ensemble. <math>A \\cap A =\\{x/((x \\in A) \\, et \\, (x \\in A))\\}=\\{x/(x \\in A) \\}=A</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:De m\u00eame pour la r\u00e9union</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soient A et B deux ensembles. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>A \\cap B = \\{x/((x \\in A)\\, et \\, (x \\in B))\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:::<math>=\\{x/((x \\in B)\\, et \\, (x \\in A))\\}=B \\cap A</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:De m\u00eame pour la r\u00e9union</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soient A, B, C trois ensembles</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>A \\cap B=\\{x/(x \\in A)\\, et\\, (x \\in B)\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>(A \\cap B)\\cap C=\\{x/((x \\in A)\\, et\\, (x \\in B))\\, et \\, (x \\in C) \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::::<math>=\\{x/(x \\in A)\\, et\\, ((x \\in B)\\, et \\, (x \\in C)) \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>=A \\cap (B \\cap C)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:De m\u00eame pour la r\u00e9union</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Propositions :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soit ''E'' un ensemble quelconque.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Double passage au compl\u00e9mentaire : <br/><math>\\forall A\\in \\mathcal P(E), \\complement_E\\left(\\complement_E A\\right)=A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Lois de Morgan :<br/>Le compl\u00e9mentaire d'une r\u00e9union est l'intersection des compl\u00e9mentaires, et le compl\u00e9mentaire d'une intersection est la r\u00e9union des compl\u00e9mentaires, c'est-\u00e0-dire<br/><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), \\complement_E\\left(A\\cap B\\right)=\\left(\\complement_E A\\right) \\cup \\left(\\complement_E B\\right)</math><br/><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), \\complement_E\\left(A\\cup B\\right)=\\left(\\complement_E A\\right) \\cap \\left(\\complement_E B\\right)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), A\\cap B=\\emptyset\\Leftrightarrow A\\subset \\complement_E B</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\"><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), A\\cup B=E\\Leftrightarrow \\complement_E B\\subset A</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), A\\subset B\\Leftrightarrow \\complement_E B\\subset \\complement_E A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Propri\u00e9t\u00e9s de la diff\u00e9rence sym\u00e9trique ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Propositions :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Soit ''E'' un ensemble quelconque.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Commutativit\u00e9 de la diff\u00e9rence sym\u00e9trique </ins>:<ins class=\"diffchange diffchange-inline\"><br</ins>/<ins class=\"diffchange diffchange-inline\">><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), A\\Delta B=B\\Delta A<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<math>\\forall A\\in \\mathcal P(E), A\\Delta A=\\emptyset, A\\Delta E=\\complement_E A</math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<math>\\emptyset<</ins>/<ins class=\"diffchange diffchange-inline\">math> est \u00e9l\u00e9ment neutre:<br</ins>/<ins class=\"diffchange diffchange-inline\">><math>\\forall A\\in \\mathcal P(E), A\\Delta \\emptyset=A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Associativit\u00e9 de la diff\u00e9rence sym\u00e9trique </ins>:<ins class=\"diffchange diffchange-inline\"><br/><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), \\forall C\\in \\mathcal P(E), (A\\Delta B)\\Delta C=A\\Delta (B\\Delta C)</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Distributivit\u00e9 de <math>\\cap</math> par rapport \u00e0 <math>\\Delta</math><br/><math>\\forall A\\in \\mathcal P(E), \\forall B\\in \\mathcal P(E), \\forall C\\in \\mathcal P(E), A\\cap (B\\Delta C)=(A\\cap B) \\Delta (A\\cap C)<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Alg\u00e8bre|retour \u00e0 l'alg\u00e8bre]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Produit d'ensembles ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finitions :'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">Soient ''x'' et ''y'' deux objets. Nous appelons '''couple''' (''x'', ''y'') la suite d'objets dont le premier \u00e9l\u00e9ment est ''x'' et le second ''y''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soient ''X'' et ''Y'' deux ensembles quelconque. Nous appelons '''produit cart\u00e9sien''' ou '''produit''' de ''X'' et de ''Y'' l'ensemble des couples (''x'', ''y'') tels que ''x'' appartient \u00e0 ''X'' et ''y'' appartient \u00e0 ''Y''. Cet ensemble se note <math>X\\times Y</math>. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Formellement, le couple <math>(x,y)</math> peut \u00eatre d\u00e9fini ainsi </ins>: <ins class=\"diffchange diffchange-inline\">si <math>x<</ins>/<ins class=\"diffchange diffchange-inline\">math> et <math>y<</ins>/<ins class=\"diffchange diffchange-inline\">math> sont deux objets, alors <math>(x,y)=\\{\\{x\\},\\{x,y\\}\\}</math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Cette d\u00e9finition assure en particulier que <math>(a,b)=(c,d) \\Leftrightarrow (a=c \\,\\land\\, b=d)</math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''Remarque :''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Attention <math>(x,y)\\neq (y,x)<</ins>/<ins class=\"diffchange diffchange-inline\">math> en g\u00e9n\u00e9ral. Il ne faut surtout pas confondre un couple avec une paire <math>\\left\\{ x,y\\right\\}</math> pour laquelle nous avons <math>\\left\\{ x,y\\right\\} =\\left\\{ y,x\\right\\}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''D\u00e9finitions </ins>:<ins class=\"diffchange diffchange-inline\">'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soient ''x''<sub>1<</ins>/<ins class=\"diffchange diffchange-inline\">sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> ''n'' objets. Nous appelons ''n''-'''uplet''' (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) la suite d'objets dont le premier \u00e9l\u00e9ment est ''x''<sub>1</sub>, le deuxi\u00e8me ''x''<sub>2</sub>, ..., et le dernier \u00e9l\u00e9ment ''x''<sub>''n''</sub>. Ces \u00e9l\u00e9ments sont appel\u00e9s '''composantes'''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Soient ''E''<sub>1</sub>, ''E''<sub>2</sub>, ..., ''E''<sub>n</sub> ''n'' ensembles quelconques. Nous appelons '''produit cart\u00e9sien''' ou '''produit''' de ''E''<sub>1</sub> par ... par ''E''<sub>''n''</sub>, l'ensemble des ''n''-uplets (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) tels que ''x''<sub>1</sub> appartient \u00e0 ''E''<sub>1</sub>, ..., ''x''<sub>''n''</sub> appartient \u00e0 ''E''<sub>''n''</sub>. Cet ensemble se note <math>E_1\\times E_2\\times\\ldots\\times E_n</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Si ''E''<sub>1</sub>= ''E''<sub>2</sub>=...= ''E''<sub>''n''</sub> sont \u00e9gaux \u00e0 un m\u00eame ensemble ''E'', nous notons ''E''<sup>''n''</sup> plut\u00f4t que <math>E\\times E\\times \\ldots\\times E</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Alg\u00e8bre|retour \u00e0 l'alg\u00e8bre]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Cat\u00e9gorie</ins>:<ins class=\"diffchange diffchange-inline\">Alg\u00e8bre (livre)|Th\u00e9orie \u00e9l\u00e9mentaire des ensembles]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Cat\u00e9gorie:Alg\u00e8bre (livre)|Th\u00e9orie \u00e9l\u00e9mentaire des ensembles]</ins>]</div></td></tr>\n"
}
}