I know the General Addition Rule for Two Events is
$$P (A\cup B) = P (A) + P(B) - P(A\cap B)$$
I know the proof where ${R_1} = A\cap B, R_2 = A\cap B', R_3 = A'\cap B,$ and $R_4 = A'\cap B'.$ The $R_i$'s are a partition of a sample space S and they are disjoint.
However I was wondering if we can prove this by making use of the set equations:
$A\cup B =A\cup(A'\cap B) $(I know the RHS is disjoint) and
$B = (A\cap B)\cup(A'\cap B) $(where the RHS is disjoint).
How would one go proving the theorem using these set equations? Can someone please show me?
$A\cup B =A\cup(A'\cap B)$ (the RHS is disjoint) implies $P(A\cup B) =P(A)+P(A'\cap B)$
$B = (A\cap B)\cup(A'\cap B)$ (the RHS is disjoint) implies $P(B)=P(A\cap B)+P(A'\cap B)$
Now substitute $P(A'\cap B)$ from the second equation into the first. Is this what you are asking?