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I need to prove that

$$P(A \backslash B) = P(A) - P(A\cap B)$$


$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$

using the axioms of probability but can't see where to start. Note: this is homework, so a hint only would be nice :)

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Corrected question :) – Callum Rogers Nov 16 '10 at 17:41
up vote 3 down vote accepted

Hint: Think "countable additivity". What can you say about $A\setminus B$ and $A\cap B$ if you are hoping to use this axiom.

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The second one as written is false. I'm sure you meant

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Jonas' answer is a good hint on how to apply the first part to prove the second part.

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Let $B \subset A$, then $A = B \cup (A \setminus B)$ which is a disjoint union so...

Now, $$A \cup B = (A \setminus (A \cap B)) \cup (A \cap B) \cup (B \setminus (A \cap B))$$ which is again a disjoint union. So...

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