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