Exercice rapide de théorie des ensembles

Soit $E$ un ensemble et $A,B,C$ des parties de $E$ .

Quelle condition nécessaire et suffisante sur ces parties donne l’égalité ensembliste
$A\cup B=B\cap C\,\, ?$

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Solution by Ancient TreeEN

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First, we notice that the inclusion from right to left is automatic, since, if $x$ is in $B\cap C$ , it is in particular in $B$ , and therefore in $A\cup B$ .

For the inclusion from left to right, if $x\in A$ , $x$ has to be in $B$ and $C$ , which means that $A\subseteq B$ and $A\subseteq C$ ; and if $x\in B$ , then it is enough that $x\in C$ , which means that $B\subseteq C$ .

In total, the sufficient and necessary condition is $A \subseteq B \subseteq C$ .

Solution by Sequoia

Discussions1 useful vote

On a que $A$ est inclus dans $A\cup B$ qui vaut $B\cap C$ donc $A$ est inclus dans $B$ et $C$ . De même, $B$ est inclus dans $A\cup B=B\cap C$ qui est donc inclus dans $C$ . Ainsi $A\subset B\subset C$ . Réciproquement, si cette suite d’inclusion est vérifiée, alors $A\cup B=B$ et $B\cap C= B$ donc on a bien $A\cup B = B \cap C$ .

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