Q: (possibly?) equivalent defintiioons of indepdnt events Let $A_1$ and $A_2$ denote two events.
Now I'm sure from school we've read that $A_1$ and $A_2$ are called independent when
$P(A_1)P(A_2) = P(A \cup A_2)$
Now I've read another definition (albeit usually to do with more than 2 events) saying that $A_1$ and $A_2$ are independent when, basically, for all combinations of $A_1$ and $A_2$ happening or not happening, to work out the probability of both those events we just multiply them together. eg, for example (there are four examples to consider) we have:
$P(A_1^c \cup A_2) = P(A_1^c)P(A_2)$
My question is, are these two definitions actually equivalent (for both two events and more?)

A: $A_2 = (A_1 \cap A_2) \cup (A_1^c \cap A_2)$, while  $(A_1 \cap A_2)$  and $(A_1^c \cap A_2)$ are mutually exclusive, so $P(A_2) = P(A_1 \cap A_2)+P(A_1^c \cap A_2)$ and $P(A_1^c \cap A_2)=P(A_2) - P(A_1 \cap A_2).$
If $P(A_1)P(A_2) = P(A_1 \cap A_2)$ then $$P(A_1^c)P(A_2) = (1-P(A_1))P(A_2) = P(A_2)-P(A_1)P(A_2) = P(A_2)-P(A_1 \cap A_2) = P(A_1^c \cap A_2).$$ 
You can do something similar to get from any of the four cases to any of the others.
