if RVs $X$ and $Y$ are indicators of independent events, does that imply their complements do too? $ p(X \cap Y)=p(X)p(Y) \implies p(X^c \cap Y^c)=p(X^c)p(Y^c)?$
im having trouble deciding weither the above statement is true or not, my intuation is that its true, can any one prove or contradict it?
btw, this is not Homework so no homework tag needed.

There are a few possible questions we might try to answer. First, let's prove that if $X$ and $Y$ are independent random variables, then $1-X$ and $1-Y$ are independent.
Hint: Consider $$\mathbb{Pr}(1 - X \leq a, 1-Y \leq b) = \mathbb{Pr}(X \geq 1-a, Y \geq 1-b) = \mathbb{Pr}(X \geq 1-a)\mathbb{Pr}(Y \geq 1-b)  $$
Do you also want to prove that indicators are independent iff the events they refer to are as well?
EDIT: If $X = \mathbb{1}_A$ and $Y = \mathbb{1}_B$, then $X$ and $Y$ are independent iff $A$ and $B$ are.
Hint: If $A$ and $B$ are not independent then $Pr(A)Pr(B) \neq Pr (A \cap B)$ thus $Pr(X=1, Y=1) \neq Pr(X = 1)Pr(Y = 1)$. Conversely, you might simply consider $Pr(X \leq a, Y \leq b)$ where without loss of generality we might take $a$ and $b$ to be  $0$ or $1$.
