Cardinality of sets of functions with well-ordered domain and codomain I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated.
If $X$ and $Y$ are well-ordered sets, then determine the cardinality of:


*

*$\{f : f$ is a function from $X$ to $Y\}$

*$\{f : f$ is an order-preserving function from $X$ to $Y\}$

*$\{f : f$ is a surjective and order-preserving function from $X$ to $Y\}$


*

*The cardinality of the set of functions from $X$ to $Y$ is the definition of the cardinal $Y^X$. 

*The number of order-preserving functions from $X$ to $Y$, given that well-orders of each set have been fixed, depends on the nature of those orders. For example, there are no such orders in the case that the order type of $X$ is longer than the order type of $Y$. If  $X$ and $Y$ are finite, then there is some interesting combinatorics involved to give the right answer. For example, if both are finite of the same size, there is only one order-preserving function. If $Y$ is one bigger, then there are $Y$ many (you can put the hole anywhere). And so on. If $Y$ is infinite, of size at least $X$, then you get $Y^X$ again, since you can code any function into the omitted part, by leaving gaps of a certain length. 

*A surjective order-preserving map is an isomorphism, and for well-orders, this is unique if it exists at all, so the answer is either 0 or 1, depending on whether the orders are isomorphic or not.
