Addition Law of Probability/Proof 2

Proof
From Set Difference and Intersection form Partition:


 * $A$ is the union of the two disjoint sets $A \setminus B$ and $A \cap B$
 * $B$ is the union of the two disjoint sets $B \setminus A$ and $A \cap B$.

So, by the definition of probability measure:
 * $\map \Pr A = \map \Pr {A \setminus B} + \map \Pr {A \cap B}$
 * $\map \Pr B = \map \Pr {B \setminus A} + \map \Pr {A \cap B}$

From Set Difference is Disjoint with Reverse:
 * $\paren {A \setminus B} \cap \paren {B \setminus A} = \O$

Hence:

Hence the result.