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Let $S$ be a discrete sample space, and let $X:S\to\nats$ and
$Y:S\to\nats$ be two random variables.

\bparts
\ppart
Let $A$ and $B$ be two subsets of natural numbers. Write $(X\in A)$
for the event that $X$ assumes a value in $A$, and likewise $(Y\in
B)$.

Prove that if $X$ and $Y$ are independent random variables, then
$(X\in A)$ and $(Y\in B)$ are independent events.

\ppart
If $g, f:\nats\to\nats$ are functions, $f(X)$ and $g(Y)$ are also
random variables on $S$.

Prove that if $X$ and $Y$ are independent, then $f(X)$ and
$g(Y)$ are independent as well.

Hint: use part (a).

\eparts

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