Exercice :
Soient \(E\) un ensemble et \(A\), \(B\) et \(C\) trois parties de \(E\).
Montrer que :
\[
(A \cup B) \cap (B \cup C) \cap (C \cup A)
= (A \cap B) \cup (B \cap C) \cup (C \cap A).
\]
Corrigé :
On a :
\[
\begin{aligned}
&(A \cup B) \cap (B \cup C) \cap (C \cup A) \\
&= [(A \cup B) \cap (C \cup B)] \cap (C \cup A) \\
&= [(A \cap C) \cup B] \cap (C \cup A) \\
&= [(A \cap C) \cap C] \cup [(A \cap C) \cap A] \\
&\qquad \cup (B \cap C) \cup (B \cap A) \\
&= (A \cap C) \cup (A \cap C) \\
&\qquad \cup (B \cap C) \cup (A \cap B) \\
&= (A \cap C) \cup (B \cap C) \cup (A \cap B).
\end{aligned}
\]
Remarque :
On a \(
(A \cap C) \cap A = A \cap C,
\) car \(A \cap C \subset A\).