The set of function being countable?

Question

I am confused with the following notation for the set:
$$\{\ f:A \to B\}$$ How such a set is interpreted? Does this even make sense to refer to this as a set? Is it a set whose member is a function? And how can we make any claim about this set being countable or uncountable? I am not referring to any specific sets $A$ and $B$, but I am looking for some instructive ideas.

Answer 1

My favorite notation for this is $Y^X$. See functional analysis.

As mentioned by @Henning Makholm, the space is uncountable as soon as $X$ is infinite and $Y$ has at least $2$ elements. This follows from Cantor's diagonal argument.