Given two sets and , let be the set of all functions . Prove that for any set , .
Here, . In other words, we want to prove that ?
So is the set of all functions . It does not say anything about the sets being countable/uncountable, or finite/infinite. Since any element maps to either of two values, then using the characteristic function, the total number of functions is if is countably infinite or finite. But if is uncountable/infinite, then ?