# Independent Events are Independent of Complement/Corollary

## Corollary to Independent Events are Independent of Complement

Let $A$ and $B$ be events in a probability space $\struct {\Omega, \Sigma, \Pr}$.

$A$ and $B$ are independent if and only if $\Omega \setminus A$ and $\Omega \setminus B$ are independent.

## Proof

Let $A$ and $B$ be independent.

Then from Independent Events are Independent of Complement, $A$ and $\Omega \setminus B$ are independent.

Setting $A' = \Omega \setminus B$ and $B' = A$, we see clearly that $A'$ and $B'$ are independent.

So from the main result, $A'$ and $\Omega \setminus B'$ are independent.

That is, $\Omega \setminus B$ and $\Omega \setminus A$ are independent.

The "only if" part of the result follows directly from Relative Complement of Relative Complement and another application of this result.

$\blacksquare$