(i) $Y^\emptyset$ has exactly one element, namely $\emptyset$, whether $Y$ is empty or not, and (ii) if $X$ is not empty, then $\emptyset^X$ is empty.
How do you prove these statements to be true?
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$X^Y$ is the set of functions with domain $Y$ and taking values in $X$. You should be able to prove these two statements now that you have the definition. It makes no sense to ask for help before you think about the problems. – Andrés E. Caicedo Aug 17 '18 at 20:25
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That notation stands for the set of functions with domain equal $Y$ and range a subset of $X$. – Ned Aug 17 '18 at 20:28