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Can someone tell me where the error in my calculation is?

$|A \cup B \cup C|$

$|(A \cup B) \cup C|$

$|A \cup B| + |C| - |(A \cup B) \cap C|$

$|A \cup B| + |C| - |A \cup (B \cap C)|$

$|A| + |B| - |A \cap B| + |C| - |A| - |B \cap C| + |A \cap B \cap C|$

Rearranged gives:

$|A| + |B| + |C| - |A \cap B| - |B \cap C| + |A \cap B \cap C| - |A|$

This is the answer that I'm supposed to get as opposed to the one above

$|A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

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3 Answers

$(A \cup B)\cap C$ is not the same thing as $A \cup (B \cap C)$.

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$$((A \cup B) \cap C)\neq(A \cup (B \cap C))$$ rather it is equal to $$((A \cap C) \cup (B \cap C))$$

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$(A \cup B)\cap C=(A\cap C)\cup (B\cap C)$

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