La théorie des ensembles est une branche des mathématiques qui étudie les ensembles. Elle s'intéresse à ces regroupements d'éléments et aux manières de les considérer comme des objets mathématiques.
Ce qu'elle étudie
Cette discipline examine les ensembles ainsi que les sous-ensembles qu'ils contiennent. Un sous-ensemble est un ensemble formé d'éléments pris dans un autre ensemble. L'étude de ces regroupements donne un langage pour raisonner sur les collections d'objets et sur leurs relations d'inclusion.
$$|\mathcal{P}(E)| = 2^{|E|}$$
L'égalité indique qu'un ensemble ayant n éléments possède deux puissance n sous-ensembles.
À quoi elle sert
Elle sert à exprimer clairement des relations entre collections d'objets, notamment :
- distinguer un ensemble de ses sous-ensembles ;
- étudier les inclusions entre ensembles ;
- compter les sous-ensembles d'un ensemble donné.
Où elle se place
Elle est une branche des mathématiques consacrée à l'étude des ensembles.
Deux infinis
La grille montre comment chaque couple d'entiers naturels \((i, j)\) reçoit un numéro en parcourant successivement les diagonales. Cette numérotation prouve que \(\mathbb{N} \times \mathbb{N}\) est dénombrable, donc que les rationnels le sont aussi ; à l'opposé, l'argument diagonal établit que les réels ne le sont pas. La figure met ainsi en évidence deux infinis différents, au cœur de la théorie de Cantor.
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