گام به گام اثبات رو انجام میدیم . امیداورم از طولانی بودن آن خسته نشید :)
مرحله اول:
$$( A \Delta B)\Delta C =$$
$$ \{[(A \Delta B)] \cup C\}- \{[(A \Delta B)] \cap C\}=$$
$$ \{[(A \Delta B)] \cup C\}\cup \{[(A \Delta B)] \cap C\}'=$$
$$\{[(A \cup B) \cap (A' \cup B')] \cup C\} \cap \{[(A \cup B) \cap (A' \cup B')] \cap C\}'=$$
$$\{[(A \cup B) \cap (A' \cup B')] \cup C\} \cap \{[(A \cup B) \cap (A' \cup B')]' \cap C'\}=$$
$$(A \cup B \cup C) \cap (A' \cup B' \cup C) \cap \{[(A \cup B )' \cup (A' \cup B')]' \cup C'\}=$$
$$(A \cup B \cup C) \cap (A' \cup B' \cup C) \cap \{[(A' \cap B' ) \cup (A \cap B)] \cup C'\}=$$
$$(A \cup B \cup C) \cap (A' \cup B' \cup C) \cap \{[(A' \cap B' ) \cup A] \cap [ (A '\cap B') \cup B] \cup C'\}$$
$$=(A \cup B \cup C) \cap (A' \cup B' \cup C) \cap \{[(B' \cup A) \cap (A' \cup B)] \cup C'\}$$
$$=(A \cup B \cup C) \cap (A' \cup B' \cup C) \cap (A \cup B' \cup C') \cap ( A'\cup C' \cup B )$$
مرحله دو:
$$A\Delta (B \Delta C)= $$
$$A \Delta [(B \cup C )-( B\cap C)] =$$
$$A \Delta [(B \cup C ) \cap ( B\cap C)'] =$$
$$A \Delta [(B \cup C ) \cap ( B'\cap C')]= $$
$$\{ A\cup [(B \cup C)(B' \cup C')]\}- \{ A \cap [(B \cup C) \cap (B' \cup C')]\} =$$
$$\{ A\cup [(B \cup C)(B' \cup C')]\} \cap \{ A \cap [(B \cup C) \cap (B' \cup C')]\}' =$$
$$\{ A\cup [(B \cup C) \cap (B' \cup C')]\}\cap \{ A' \cup [(B \cup C) \cap (B' \cup C')]\}' =$$
$$\{ A\cup [(B \cup C)(B' \cup C')]\}\cap \{ A \cap [(B \cup C) \cap (B' \cup C')]\}' =$$
$$(A \cup B \cup C) \cap ( A\cup B' \cap C' )\cup \{ A \cap [(B' \cap C') \cap (B \cap C)]\}' =$$
$$(A \cup B \cup C) \cap ( A\cup B' \cup C' )\cap \{ A' \cup [B' \cap C' \cup B] \cap [(B' \cap C') \cup C]\} $$
$$=(A \cup B \cup C) \cap ( A\cup B' \cup C' )\cap \{A' \cup [(C' \cup B) \cap (B' \cup C)] \}$$
$$=(A \cup B \cup C) \cap ( A\cup B' \cup C' )\cap ( A'\cup C' \cup B) \cap ( A'\cup B' \cup C )$$
مرحله یک و دو رو نیگا :
$$(A \Delta B)\Delta C=A\Delta(B \Delta C)$$
پایان:)
