{"id":309234,"date":"2026-10-27T14:48:52","date_gmt":"2026-10-27T13:48:52","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=309234"},"modified":"2026-10-27T14:51:53","modified_gmt":"2026-10-27T13:51:53","slug":"theorie-des-ensembles","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/theorie-des-ensembles\/","title":{"rendered":"Th\u00e9orie des ensembles : d\u00e9finition et op\u00e9rations expliqu\u00e9es"},"content":{"rendered":"<section class=\"you-know\"><div class=\"you-know__title\"><p>\ud83e\udde0 \u00c0 retenir :<\/p><\/div><div class=\"you-know__text\"><ul>\n<li>La th\u00e9orie des ensembles analyse des collections d\u2019objets, ou ensembles, avec des notations et symboles propres.<\/li>\n<li>Les op\u00e9rations comme l\u2019union, l\u2019intersection et la diff\u00e9rence cr\u00e9ent de nouveaux ensembles ou comparent leurs \u00e9l\u00e9ments.<\/li>\n<li>Elle constitue le fondement de plusieurs domaines comme l\u2019informatique, la logique et les probabilit\u00e9s.<\/li>\n<\/ul><\/div><\/section>\n\n<p>La <strong>th\u00e9orie des ensembles<\/strong> est une branche fondamentale des <a href=\"https:\/\/sherpas.com\/cours\/maths\">math\u00e9matiques<\/a> qui traite de la collection d\u2019objets appel\u00e9s \u00ab\u00a0ensembles\u00a0\u00bb. Cette \u00e9tude permet aux math\u00e9maticiens de comprendre les <strong>relations<\/strong> entre diff\u00e9rents groupes d\u2019\u00e9l\u00e9ments, de d\u00e9crire leurs propri\u00e9t\u00e9s et d\u2019analyser leur structure. La connaissance de cette th\u00e9orie est essentielle pour appr\u00e9hender des concepts plus complexes en math\u00e9matiques.<\/p>\n\n<section class=\"you-know\">\n                <div class=\"you-know__title\"><p>\ud83d\udccc En bref<\/p>\n<\/div>\n                        <div class=\"you-know__text\">\n            <p>La th\u00e9orie des ensembles \u00e9tudie les ensembles, collections d\u2019objets distincts appel\u00e9s \u00e9l\u00e9ments. Les symboles de base sont \u2208 (appartient \u00e0), \u2282 (est inclus dans) et \u2205 (ensemble vide). Un ensemble se d\u00e9crit en extension ou par sa propri\u00e9t\u00e9 caract\u00e9ristique. Ses op\u00e9rations sont l\u2019union, l\u2019intersection, la diff\u00e9rence A\u2216B et le compl\u00e9mentaire. Fond\u00e9e par Georg Cantor d\u00e8s 1874, la th\u00e9orie devient axiomatique avec Zermelo (1908) et Fraenkel (1922).<\/p>\n\n        <\/div>\n              <\/section>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"quest-ce-quun-ensemble\">Qu\u2019est-ce qu\u2019un ensemble ?<\/h2>\n\n\n<p>Un <strong>ensemble<\/strong> est une collection bien d\u00e9finie d\u2019objets distincts. Ces objets peuvent \u00eatre des nombres, des lettres, ou m\u00eame d\u2019autres ensembles. En g\u00e9n\u00e9ral, les \u00e9l\u00e9ments d\u2019un ensemble sont list\u00e9s entre accolades et s\u00e9par\u00e9s par des virgules.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"exemple-densemble\">Exemple d\u2019ensemble<\/h3>\n\n\n<p>Par exemple, l\u2019ensemble des cinq premiers entiers naturels non nuls peut s\u2019\u00e9crire comme suit : {1, 2, 3, 4, 5}.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"decrire-un-ensemble-par-sa-propriete-caracteristique\">D\u00e9crire un ensemble par sa propri\u00e9t\u00e9 caract\u00e9ristique<\/h3>\n\n\n\n<p>D\u00e9crire un ensemble par sa propri\u00e9t\u00e9 caract\u00e9ristique revient \u00e0 donner la condition que v\u00e9rifient tous ses \u00e9l\u00e9ments, et eux seuls : c\u2019est la d\u00e9finition en compr\u00e9hension. L\u2019ensemble {0, 2, 4, 6, 8}, \u00e9crit en extension, devient ainsi {x \u2208 \u2115 | x pair et x &lt; 10}.<\/p>\n\n\n<section class=\"push \">\n        <div >\n            \n                                                <div class=\"push__container  qcd\" data-target=\"_self\" data-qcd=\"aHR0cHM6Ly9zaGVycGFzLmNvbS9mdW5uZWwtc3R1ZGVudC1xdWFsaWZpY2F0aW9u\" data-banner=\"2\" data-name=\"CPM1 Maths\" style=\"background-color: #F3FBF7\" >\n                                                                        <div class=\"push__teachers\">\n    <div class=\"d-flex justify-content-center d-md-none\">\n        <img class=\"push__logo\" src=\"\/content\/themes\/sherpas-theme\/dist\/images\/logo.svg\" alt=\"Logo Sherpas\">\n    <\/div>\n    <div class=\"swiper\">\n        <div class=\"swiper-wrapper\">\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                          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class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                 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xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">20\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Cle\u0301mence2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Cl\u00e9mence<\/p>                          <p class=\"push__teachers-school\">HEC Paris<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                            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          class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Bastien2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Bastien<\/p>                          <p class=\"push__teachers-school\">Polytechnique<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            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xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">26\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Pierre2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Pierre<\/p>                          <p class=\"push__teachers-school\">ESSEC<\/p>                                                     <div 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  class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Simon2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Simon<\/p>                          <p class=\"push__teachers-school\">4e ann\u00e9e de m\u00e9decine<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            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xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">26\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Jade2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Jade<\/p>                          <p class=\"push__teachers-school\">Sciences Po Paris<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                 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       <div  class=\"d-none d-md-block\" >\n                                    <img class=\"push__logo\" src=\"\/content\/themes\/sherpas-theme\/dist\/images\/logo.svg\"\n                                        alt=\"Logo Sherpas\">\n                                <\/div>\n                                                                                        <div class=\"push__title\"><p>Besoin d&rsquo;un <span style=\"color: #4cc583;\">prof particulier<\/span> de maths ? <span role=\"image\" aria-label=\"\u2728\">\u2728<\/span><\/p>\n<\/div>\n                                                                                        <div class=\"push__text\"><p>Nos Sherpas sont l\u00e0 pour t&rsquo;aider \u00e0 progresser et prendre confiance en toi !<\/p>\n<\/div>\n                                                        <div class=\"push__promo-button   \">\n                                                                                                    <div\n                                        class=\"btn btn-primary\"\n                                        style=\"color: ; border-color: ; background-color: \"\n\n                                    >\n                                        JE PRENDS UN COURS OFFERT !\n                                    <\/div>\n                                                            <\/div>\n                                                    <\/div>\n                                    <\/div>\n                                    <\/div>\n    <\/section>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"notation-des-ensembles\">Notation des ensembles<\/h3>\n\n\n<p>La notation des ensembles utilise divers symboles. Voici quelques-uns des symboles les plus couramment utilis\u00e9s :<\/p>\n\n<ul>\n    <li>{ } &#8211; Accolades utilis\u00e9es pour d\u00e9finir un ensemble<\/li>\n    <li><math><mo>&#x2208;<\/mo><\/math> &#8211; Symbolise que quelque chose appartient \u00e0 un ensemble (par exemple, <math><mi>a<\/mi><mo>&#x2208;<\/mo><mi>A<\/mi><\/math> signifie \u00ab\u00a0a appartient \u00e0 l\u2019ensemble A\u00a0\u00bb)<\/li>\n    <li><math><mo>&#x2209;<\/mo><\/math> &#8211; Signifie que quelque chose n\u2019appartient pas \u00e0 un ensemble <\/li>\n    <li><math><mo>&#x2282;<\/mo><\/math> &#8211; Indique qu\u2019un ensemble est un sous-ensemble d\u2019un autre (<math><mi>A<\/mi><mo>&#x2282;<\/mo><mi>B<\/mi><\/math> signifie \u00ab\u00a0A est un sous-ensemble de B\u00a0\u00bb)<\/li>\n    <li><math><mo>&#x222A;<\/mo><\/math> &#8211; Union de deux ensembles<\/li>\n    <li><math><mo>&#x2229;<\/mo><\/math> &#8211; Intersection de deux ensembles<\/li>\n  <\/ul>\n\n<div style=\"position:relative;overflow:hidden;padding-top:56.25%\"><iframe style=\"position:absolute;top:0;left:0;width:95%;height:100%;border:0\" src=\"https:\/\/www.youtube.com\/embed\/n-K61k0RxxM?si=nhwHQ6IwZmwJM1by&#038;controls=0\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"types-densembles\">Types d\u2019ensembles<\/h2>\n\n\n<p>Il existe plusieurs types d\u2019ensembles qui jouent des r\u00f4les diff\u00e9rents en math\u00e9matiques.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"ensembles-finis-et-infinis\"><strong>Ensembles finis et infinis<\/strong><\/h3>\n\n\n<p>Un ensemble fini contient un nombre d\u00e9termin\u00e9 d\u2019\u00e9l\u00e9ments. Par exemple, {a, b, c} est un ensemble fini avec trois \u00e9l\u00e9ments. En revanche, un <strong>ensemble infini<\/strong> poss\u00e8de un nombre illimit\u00e9 d\u2019\u00e9l\u00e9ments, tel que l\u2019ensemble des <strong><a href=\"https:\/\/sherpas.com\/blog\/nombres-entiers-nombres-decimaux\/\">entiers naturels<\/a><\/strong> \u2115 = {0, 1, 2, 3, \u2026}.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"ensembles-vides-et-universels\"><strong>Ensembles vides et universels<\/strong><\/h3>\n\n\n<p>L\u2019ensemble vide, not\u00e9 <math><mo>&#x2205;<\/mo><\/math>, ne contient aucun \u00e9l\u00e9ment. \u00c0 l\u2019oppos\u00e9, l\u2019 <strong>ensemble universel<\/strong> englobe tous les \u00e9l\u00e9ments possibles dans un contexte donn\u00e9. Par exemple, l\u2019ensemble universel U pourrait \u00eatre l\u2019ensemble de tous les nombres r\u00e9els <math><mi>&#x211D;<\/mi><\/math>.<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"operations-sur-les-ensembles\">Op\u00e9rations sur les ensembles<\/h2>\n\n\n<p>Diverses op\u00e9rations permettent de manipuler les ensembles pour cr\u00e9er de nouvelles collections ou analyser des relations sp\u00e9cifiques entre eux.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"union-densembles\"><strong>Union d\u2019ensembles<\/strong><\/h3>\n\n\n<p>L\u2019<strong>union<\/strong> de deux ensembles A et B, not\u00e9e <math><mi>A<\/mi><mo>&#x222A;<\/mo><mi>B<\/mi><\/math>, est l\u2019ensemble de tous les \u00e9l\u00e9ments qui appartiennent soit \u00e0 A, soit \u00e0 B, soit aux deux. Par exemple, si <math><mi>A<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>}<\/mo><\/math> et <math><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>5<\/mn><mo>}<\/mo><\/math>, alors <math><mi>A<\/mi><mo>&#x222A;<\/mo><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>5<\/mn><mo>}<\/mo><\/math>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"intersection-densembles\"><strong>Intersection d\u2019ensembles<\/strong><\/h3>\n\n\n<p>L\u2019<strong>intersection<\/strong> de deux ensembles A et B, not\u00e9e <math><mi>A<\/mi><mo>&#x2229;<\/mo><mi>B<\/mi><\/math>, est l\u2019ensemble de tous les \u00e9l\u00e9ments qui appartiennent \u00e0 la fois \u00e0 A et \u00e0 B. Si <math><mi>A<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>}<\/mo><\/math> et <math><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>5<\/mn><mo>}<\/mo><\/math>, alors <math><mi>A<\/mi><mo>&#x2229;<\/mo><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>3<\/mn><mo>}<\/mo><\/math>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"difference-densembles\"><strong>Diff\u00e9rence d\u2019ensembles<\/strong><\/h3>\n\n\n<p>La <strong>diff\u00e9rence<\/strong> entre deux ensembles A et B, not\u00e9e <math><mi>A<\/mi><mo>&#x2216;<\/mo><mi>B<\/mi><\/math>, est l\u2019ensemble de tous les \u00e9l\u00e9ments qui appartiennent \u00e0 A mais pas \u00e0 B ; on lit \u00ab A priv\u00e9 de B \u00bb. Par exemple, si <math><mi>A<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>}<\/mo><\/math> et <math><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>5<\/mn><mo>}<\/mo><\/math>, alors <math><mi>A<\/mi><mo>&#x2216;<\/mo><mi>B<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>}<\/mo><\/math>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"complementation-densembles\"><strong>Compl\u00e9mentation d\u2019ensembles<\/strong><\/h3>\n\n\n<p>Le <strong>compl\u00e9ment<\/strong> d\u2019un ensemble A, souvent not\u00e9 <math><mi>A<\/mi><mo>\u2019<\/mo><\/math>, est constitu\u00e9 de tous les \u00e9l\u00e9ments qui n\u2019appartiennent pas \u00e0 A. Si l\u2019univers du discours est l\u2019ensemble des entiers relatifs <math><mi>&#x2124;<\/mi><\/math>, et <math><mi>A<\/mi><mo>=<\/mo><mo>{<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>}<\/mo><\/math>, alors le compl\u00e9ment de A dans <math><mi>&#x2124;<\/mi><\/math> serait <math><mi>A<\/mi><mo>\u2019<\/mo><mo>=<\/mo><mo>{<\/mo><mo>&#8230;<\/mo><mo>,<\/mo><mo>&#8211;<\/mo><mn>3<\/mn><mo>,<\/mo><mo>&#8211;<\/mo><mn>2<\/mn><mo>,<\/mo><mo>&#8211;<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>5<\/mn><mo>,<\/mo><mo>&#8230;<\/mo><mo>}<\/mo><\/math>.<\/p>\n\n<div style=\"position:relative;overflow:hidden;padding-top:56.25%\"><iframe style=\"position:absolute;top:0;left:0;width:95%;height:100%;border:0\" src=\"https:\/\/www.youtube.com\/embed\/bPT6-g3B5wQ?si=gYznWqtquG6rRfFb&#038;controls=0\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"roles-et-applications-de-la-theorie-des-ensembles\"><strong>R\u00f4les et applications de la th\u00e9orie des ensembles<\/strong><\/h2>\n\n\n<p>La th\u00e9orie des ensembles joue un r\u00f4le crucial dans diff\u00e9rentes branches des math\u00e9matiques et trouve des applications vari\u00e9es dans les <strong>sciences<\/strong> et les <strong>technologies<\/strong>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"fondements-des-mathematiques\"><strong>Fondements des math\u00e9matiques<\/strong><\/h3>\n\n\n<p>Les principes de la th\u00e9orie des ensembles servent de base \u00e0 divers domaines math\u00e9matiques tels que l\u2019analyse, l\u2019alg\u00e8bre et la topologie. Il est compliqu\u00e9 d\u2019imaginer ces domaines sans les concepts fondamentaux extraits de la th\u00e9orie des ensembles.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"informatique-theorique\">Informatique th\u00e9orique<\/h3>\n\n\n<p>En informatique, la th\u00e9orie des ensembles est utilis\u00e9e pour g\u00e9rer les bases de donn\u00e9es, d\u00e9velopper des algorithmes et formaliser des langages de programmation. Des structures telles que les listes et les tableaux en codage trouvent leurs racines dans les ensembles.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"logique-et-probabilite\">Logique et probabilit\u00e9<\/h3>\n\n\n<p>Dans les \u00e9tudes logiques, la th\u00e9orie des ensembles aide \u00e0 formuler et r\u00e9soudre des propositions logiques. De mani\u00e8re similaire, elle joue un r\u00f4le cl\u00e9 dans la <a href=\"https:\/\/sherpas.com\/blog\/probabilites-maths\/\">th\u00e9orie des probabilit\u00e9s<\/a> o\u00f9 les \u00e9v\u00e9nements sont souvent mod\u00e9lis\u00e9s comme des ensembles.<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"proprietes-remarquables-des-ensembles\">Propri\u00e9t\u00e9s remarquables des ensembles<\/h2>\n\n\n<p>Les ensembles poss\u00e8dent des propri\u00e9t\u00e9s sp\u00e9cifiques qui facilitent l\u2019analyse et la r\u00e9solution de probl\u00e8mes math\u00e9matiques avanc\u00e9s.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"idempotence\">Idempotence<\/h3>\n\n\n<p>La propri\u00e9t\u00e9 d\u2019 <strong>idempotence<\/strong> stipule que l\u2019union d\u2019un ensemble avec lui-m\u00eame donne cet ensemble : <math><mi>A<\/mi><mo>&#x222A;<\/mo><mi>A<\/mi><mo>=<\/mo><mi>A<\/mi><\/math>. De fa\u00e7on similaire, <math><mi>A<\/mi><mo>&#x2229;<\/mo><mi>A<\/mi><mo>=<\/mo><mi>A<\/mi><\/math>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"associativite\">Associativit\u00e9<\/h3>\n\n\n<p>L\u2019 <strong>associativit\u00e9<\/strong> permet de regrouper les ensembles sans modifier le r\u00e9sultat des op\u00e9rations. Pour l\u2019union, <math><mo>(<\/mo><mi>A<\/mi><mo>&#x222A;<\/mo><mi>B<\/mi><mo>)<\/mo><mo>&#x222A;<\/mo><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mo>&#x222A;<\/mo><mo>(<\/mo><mi>B<\/mi><mo>&#x222A;<\/mo><mi>C<\/mi><mo>)<\/mo><\/math>. Pour l\u2019intersection, <math><mo>(<\/mo><mi>A<\/mi><mo>&#x2229;<\/mo><mi>B<\/mi><mo>)<\/mo><mo>&#x2229;<\/mo><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mo>&#x2229;<\/mo><mo>(<\/mo><mi>B<\/mi><mo>&#x2229;<\/mo><mi>C<\/mi><mo>)<\/mo><\/math>.<\/p>\n\n<section class=\"push \">\n        <div >\n            \n                                                <div class=\"push__container  qcd\" data-target=\"_self\" data-qcd=\"aHR0cHM6Ly9zaGVycGFzLmNvbS9mdW5uZWwtc3R1ZGVudC1xdWFsaWZpY2F0aW9u\" data-banner=\"2\" data-name=\"CPM2 Maths\" style=\"background-color: #F3FBF7\" >\n                                                                        <div class=\"push__teachers\">\n    <div class=\"d-flex justify-content-center d-md-none\">\n        <img class=\"push__logo\" src=\"\/content\/themes\/sherpas-theme\/dist\/images\/logo.svg\" alt=\"Logo Sherpas\">\n    <\/div>\n    <div class=\"swiper\">\n        <div class=\"swiper-wrapper\">\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Emma2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Emma<\/p>                          <p class=\"push__teachers-school\">Dauphine<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">15\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Victor2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Victor<\/p>                          <p class=\"push__teachers-school\">ESCP<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">25\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Antoine2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Antoine<\/p>                          <p class=\"push__teachers-school\">Sciences Po Paris<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">18\u20ac\/h\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/David2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">David<\/p>                          <p class=\"push__teachers-school\">EDHEC<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">25\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Noemie2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">No\u00e9mie<\/p>                          <p class=\"push__teachers-school\">M2 en droit \u00e0 Assas<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">19\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Jade2x.jpg);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Jade<\/p>                          <p class=\"push__teachers-school\">Sciences Po Paris<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">21\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Fabien2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Fabien<\/p>                          <p class=\"push__teachers-school\">T\u00e9l\u00e9com Paris<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">20\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                                            <div class=\"swiper-slide\">\n                    <div class=\"push__teachers-card\">\n                        <div\n                            class=\"push__teachers-profil\"\n                            \n                            style=\"background-image: url(https:\/\/sherpas.com\/content\/uploads\/2022\/01\/Louise2x.png);\"\n                        >\n                    <\/div>\n                         <p class=\"push__teachers-name\">Louise<\/p>                          <p class=\"push__teachers-school\">Mines ParisTech<\/p>                                                     <div class=\"push__teachers-note\">\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                                    <svg  class=\"active\" >\n                                        <use\n                                            xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#star\"\/>\n                                    <\/svg>\n                                                            <\/div>\n                                                 <p class=\"push__teachers-price\">24\u20ac\/h<\/p>                     <\/div>\n                <\/div>\n                    <\/div>\n    <\/div>\n<\/div>\n                                                    <div class=\"push__content\">\n                                                            <div  class=\"d-none d-md-block\" >\n                                    <img class=\"push__logo\" src=\"\/content\/themes\/sherpas-theme\/dist\/images\/logo.svg\"\n                                        alt=\"Logo Sherpas\">\n                                <\/div>\n                                                                                        <div class=\"push__title\"><p>Ton premier <span style=\"color: #4cc483;\">cours particulier<\/span> de maths est <span style=\"color: #4cc483;\">offert<\/span> ! <span role=\"image\" aria-label=\"\ud83c\udf81\">\ud83c\udf81<\/span><\/p>\n<\/div>\n                                                                                        <div class=\"push__text\"><p>Tous nos profs sont pass\u00e9s par les meilleures \u00e9coles de France !<\/p>\n<\/div>\n                                                        <div class=\"push__promo-button   \">\n                                                                                                    <div\n                                        class=\"btn btn-primary\"\n                                        style=\"color: ; border-color: ; background-color: \"\n\n                                    >\n                                        J\u2019EN PROFITE !\n                                    <\/div>\n                                                            <\/div>\n                                                    <\/div>\n                                    <\/div>\n                                    <\/div>\n    <\/section>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"commutativite\">Commutativit\u00e9<\/h3>\n\n\n<p>Cette propri\u00e9t\u00e9 indique que l\u2019ordre des ensembles n\u2019affecte pas le r\u00e9sultat de leur union ou intersection. Ainsi, <math><mi>A<\/mi><mo>&#x222A;<\/mo><mi>B<\/mi><mo>=<\/mo><mi>B<\/mi><mo>&#x222A;<\/mo><mi>A<\/mi><\/math> et <math><mi>A<\/mi><mo>&#x2229;<\/mo><mi>B<\/mi><mo>=<\/mo><mi>B<\/mi><mo>&#x2229;<\/mo><mi>A<\/mi><\/math>.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"distributivite\">Distributivit\u00e9<\/h3>\n\n\n<p>C\u2019est la capacit\u00e9 d\u2019une op\u00e9ration \u00e0 se distribuer sur l\u2019autre. Exemple de la <strong>distributivit\u00e9<\/strong> de l\u2019union par rapport \u00e0 l\u2019intersection : <math><mi>A<\/mi><mo>&#x222A;<\/mo><mo>(<\/mo><mi>B<\/mi><mo>&#x2229;<\/mo><mi>C<\/mi><mo>)<\/mo><mo>=<\/mo><mo>(<\/mo><mi>A<\/mi><mo>&#x222A;<\/mo><mi>B<\/mi><mo>)<\/mo><mo>&#x2229;<\/mo><mo>(<\/mo><mi>A<\/mi><mo>&#x222A;<\/mo><mi>C<\/mi><mo>)<\/mo><\/math>. Et vice versa pour l\u2019intersection par rapport \u00e0 l\u2019union : <math><mi>A<\/mi><mo>&#x2229;<\/mo><mo>(<\/mo><mi>B<\/mi><mo>&#x222A;<\/mo><mi>C<\/mi><mo>)<\/mo><mo>=<\/mo><mo>(<\/mo><mi>A<\/mi><mo>&#x2229;<\/mo><mi>B<\/mi><mo>)<\/mo><mo>&#x222A;<\/mo><mo>(<\/mo><mi>A<\/mi><mo>&#x2229;<\/mo><mi>C<\/mi><mo>)<\/mo><\/math>.<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"paradoxes-et-limitations\">Paradoxes et limitations<\/h2>\n\n\n<p>Bien que la th\u00e9orie des ensembles soit robuste, elle n\u2019est pas exempte de paradoxes et de limitations qu\u2019il est int\u00e9ressant d\u2019examiner.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"paradoxe-de-russell\">Paradoxe de Russell<\/h3>\n\n\n<p>Le <strong>paradoxe de Russell<\/strong> met en lumi\u00e8re, d\u00e8s 1901, une contradiction au sein de la th\u00e9orie na\u00efve des ensembles. Posons l\u2019ensemble S des ensembles qui ne sont pas membres d\u2019eux-m\u00eames. La question devient alors : S est-il membre de lui-m\u00eame ? Cela conduit \u00e0 une contradiction si on r\u00e9pond \u00ab\u00a0oui\u00a0\u00bb ou \u00ab\u00a0non\u00a0\u00bb.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"theorie-axiomatique-des-ensembles\">Th\u00e9orie axiomatique des ensembles<\/h3>\n\n\n<p>Pour \u00e9viter ces paradoxes, la <strong>th\u00e9orie axiomatique des ensembles<\/strong> a \u00e9t\u00e9 d\u00e9velopp\u00e9e. Elle repose sur un syst\u00e8me d\u2019axiomes stricts, tels que ceux de <strong>Zermelo-Fraenkel (ZF)<\/strong>, auxquels on ajoute souvent l\u2019 <strong>axiome du choix (ZFC)<\/strong>. Ces axiomes fournissent un cadre rigoureux pour \u00e9tudier les ensembles sans contradictions. Ernst Zermelo en pose la premi\u00e8re version en 1908, compl\u00e9t\u00e9e par Abraham Fraenkel en 1922.<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"mise-en-pratique\">Mise en pratique<\/h2>\n\n\n<p>Comprendre la th\u00e9orie des ensembles n\u00e9cessite aussi sa mise en pratique \u00e0 travers des exercices et des exemples concrets. Voici quelques suggestions pour renforcer cette compr\u00e9hension :<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"exercice-1-identifier-des-ensembles\">Exercice 1 : Identifier des ensembles<\/h3>\n\n\n<p>Prenez une liste d\u2019objets autour de vous, comme des livres ou des fruits, et formez plusieurs ensembles selon des crit\u00e8res diff\u00e9rents (couleur, taille, etc.). Notez les diff\u00e9rences entre eux.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"exercice-2-operations-simples\">Exercice 2 : Op\u00e9rations simples<\/h3>\n\n\n<p>Cr\u00e9ez deux ensembles de vos connaissances amies et coll\u00e8gues. Trouvez l\u2019union, l\u2019intersection, et la diff\u00e9rence de ces ensembles et analysez ce que cela repr\u00e9sente.<\/p>\n\n\n<h3 class=\"wp-block-heading\" id=\"exercice-3-resoudre-un-paradoxe\">Exercice 3 : R\u00e9soudre un paradoxe<\/h3>\n\n\n<p>\u00c9tudiez le paradoxe de Russell en profondeur et essayez de comprendre pourquoi il cause un probl\u00e8me dans la th\u00e9orie na\u00efve des ensembles.<\/p>\n\n<div style=\"line-height:1.7;margin-top:0.75rem;margin-bottom:0.25rem;letter-spacing:-0.01em\">D\u00e9couvrez d\u2019autres <a href=\"https:\/\/sherpas.com\/blog\/concepts-statistiques-logique\/\">concepts de statistiques et de logique<\/a> :<br><ul>\n<li><a href=\"https:\/\/sherpas.com\/blog\/statistiques\/\">Statistiques<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/proportionnalite\/\">Proportionnalit\u00e9<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/logique-mathematique\/\">Logique Math\u00e9matique<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/logarithmes\/\">Logarithmes<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/combinatoire\/\">Combinatoire<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/tableau-de-variation\/\">Tableau de Variation<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/zero\/\">Z\u00e9ro en Maths<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/nombre-pi\/\">Nombre Pi<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/parents\/a\/nombres-premiers\/\">Nombres Premiers<\/a><\/li>\n<li><a href=\"https:\/\/sherpas.com\/blog\/derivees\/\">D\u00e9rivation<\/a><\/li>\n<\/ul><\/div>\n\n<section class=\"faq\">\n    <div class=\"container\" itemscope itemtype=\"http:\/\/schema.org\/FAQPage\">\n        <div class=\"faq__title\">\n                        <h2 id=\"faq\">FAQ \u2705<\/h2>\n                    <\/div>\n         <div class=\"faq__questions\">\n                                                                             <div class=\"faq__questions--item \" itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n                <div class=\"faq__questions--title  collapsed\">\n                    <p class=\"h4\" itemprop=\"name\">Comment appelle-t-on un ensemble qui a un seul \u00e9l\u00e9ment ?<\/p>\n                    <svg>\n                        <use\n                        xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#chevron-right\"\/>\n                    <\/svg>\n                <\/div>\n                <div class=\"faq__questions--content\" style=\"max-height: 0px\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                    <div itemprop=\"text\"><p>Un ensemble qui a un seul \u00e9l\u00e9ment s\u2019appelle un singleton, comme {7}. Avec deux \u00e9l\u00e9ments, comme {a, b}, on parle de paire.<\/p>\n<\/div>\n                <\/div>\n             <\/div>\n                                                                 <div class=\"faq__questions--item \" itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n                <div class=\"faq__questions--title  collapsed\">\n                    <p class=\"h4\" itemprop=\"name\">Qu\u2019est-ce que la th\u00e9orie des ensembles de Cantor ?<\/p>\n                    <svg>\n                        <use\n                        xlink:href=\"\/content\/themes\/sherpas-theme\/dist\/images\/sprite.svg#chevron-right\"\/>\n                    <\/svg>\n                <\/div>\n                <div class=\"faq__questions--content\" style=\"max-height: 0px\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                    <div itemprop=\"text\"><p>Georg Cantor fonde la th\u00e9orie dite na\u00efve des ensembles en 1874, avec un article qui montre que l\u2019ensemble infini \u211d n\u2019est pas d\u00e9nombrable, alors que l\u2019ensemble des nombres alg\u00e9briques l\u2019est, comme \u2115.<\/p>\n<\/div>\n                <\/div>\n             <\/div>\n                                <\/div>\n    <\/div>\n<\/section>\n\n\n<div class=\"kk-star-ratings kksr-auto kksr-align-center kksr-valign-bottom kksr-disabled\"\n    data-payload='{&quot;align&quot;:&quot;center&quot;,&quot;id&quot;:&quot;309234&quot;,&quot;readonly&quot;:&quot;1&quot;,&quot;slug&quot;:&quot;default&quot;,&quot;valign&quot;:&quot;bottom&quot;,&quot;ignore&quot;:&quot;&quot;,&quot;reference&quot;:&quot;auto&quot;,&quot;class&quot;:&quot;&quot;,&quot;count&quot;:&quot;1&quot;,&quot;legendonly&quot;:&quot;&quot;,&quot;score&quot;:&quot;5&quot;,&quot;starsonly&quot;:&quot;&quot;,&quot;best&quot;:&quot;5&quot;,&quot;gap&quot;:&quot;5&quot;,&quot;greet&quot;:&quot;Tu as aim\u00e9 cet article ?&quot;,&quot;legend&quot;:&quot;5\\\/5 - (1 vote)&quot;,&quot;size&quot;:&quot;24&quot;,&quot;title&quot;:&quot;Th\u00e9orie des ensembles : d\u00e9finition et op\u00e9rations expliqu\u00e9es&quot;,&quot;width&quot;:&quot;142.5&quot;,&quot;_legend&quot;:&quot;{score}\\\/{best} - ({count} {votes})&quot;,&quot;font_factor&quot;:&quot;1.25&quot;}'>\n            \n<div class=\"kksr-stars\">\n    \n<div class=\"kksr-stars-inactive\">\n            <div class=\"kksr-star\" data-star=\"1\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"2\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"3\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"4\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"5\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n    <\/div>\n    \n<div class=\"kksr-stars-active\" style=\"width: 142.5px;\">\n            <div class=\"kksr-star\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-right: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 24px; height: 24px;\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n                \n\n<div class=\"kksr-legend\" style=\"font-size: 19.2px;\">\n            5\/5 - (1 vote)    <\/div>\n    <\/div>\n","protected":false},"excerpt":{"rendered":"<p>\ud83e\udde0 \u00c0 retenir : La th\u00e9orie des ensembles analyse des collections d\u2019objets, ou ensembles, avec des notations et (&#8230;)<\/p>\n","protected":false},"author":326,"featured_media":309111,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"category":[803,810],"tag":[],"class_list":["post-309234","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/309234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/326"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=309234"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/309234\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/309111"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=309234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=309234"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=309234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}