Exercice :
Soit E un ensemble, et soient A, B, C trois parties de E.
Montrer que :
\( ( A \setminus B ) \cup ( B \setminus C ) \cup ( C \setminus A ) = ( A \cup B \cup C ) \setminus ( A \cap B \cap C ) \)
Corrigé :
On a :
\(\begin{aligned}&(A \backslash B) \cup (B \backslash C) \cup (C \backslash A)\\&= [(A \cap \bar{B}) \cup (B \cap \bar{C})] \cup (C \cap \bar{A})\\&= [(A \cup B) \cap (A \cup \bar{C}) \cap (\bar{B} \cup B) \cap (\bar{B} \cup \bar{C})] \cup (C \cap \bar{A})\\&= [(A \cup B) \cap (A \cup \bar{C}) \cap E \cap (\bar{B} \cup \bar{C})] \cup (C \cap \bar{A})\\&= [(A \cup B) \cap (A \cup \bar{C}) \cap (\bar{B} \cup \bar{C})] \cup (C \cap \bar{A})\\&= [(A \cup B) \cup C] \cap [(A \cup B) \cup \bar{A}] \cap [(A \cup \bar{C}) \cup C]\\&\quad \cap [(A \cup \bar{C}) \cup \bar{A}] \cap [(\bar{B} \cup \bar{C}) \cup C] \cap [(\bar{B} \cup \bar{C}) \cup \bar{A}]\\&= (A \cup B \cup C) \cap E \cap E \cap E \cap E \cap (\bar{A} \cup \bar{B} \cup \bar{C})\\&= (A \cup B \cup C) \cap \overline{A \cap B \cap C}\\&= (A \cup B \cup C) \backslash (A \cap B \cap C).\end{aligned}\)