Kengo Nagashima



Section 2, 問題 7(c)

Definition. \[ A \Delta B = (A \cap B^c) \cup (A^c \cap B). \] Proposition. \[ (A \Delta B) \Delta C = A \Delta (B \Delta C). \] Proof. \[ \begin{aligned} (A \Delta B) \Delta C &= [(A \cap B^c) \cup (A^c \cap B)] \Delta C \\&= [(A \cap B^c) \cup (A^c \cap B) \cap C^c] \cup [(A^c \cup B) \cap (A \cup B^c) \cap C] \\&= (A \cap B^c) \cup (A^c \cap B \cap C^c) \cup (A \cap C) \cup (B^c \cap C). \end{aligned} \] \[ \begin{aligned} A \Delta (B \Delta C) &= A \Delta [(B \cap C^c) \cup (B^c \cap C)] \\&= [A \cap (B^c \cup C) \cap (B \cup C^c)] \cup [A^c \cap (B \cap C^c) \cup (B^c \cap C)] \\&= (A \cap C) \cup (A \cap B^c) \cup (A^c \cap B \cap C^c) \cup (B^c \cap C). \end{aligned} \]

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