Proofs using the Axioms of Probability

Question

I need to prove that

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

and

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

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

Answer 1

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

Answer 2

Well,

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...

Answer 3

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.