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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:

  1. $\{f : f$ is a function from $X$ to $Y\}$
  2. $\{f : f$ is an order-preserving function from $X$ to $Y\}$
  3. $\{f : f$ is a surjective and order-preserving function from $X$ to $Y\}$
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Are you allowed to assume the axiom of choice? –  Asaf Karagila Mar 5 '11 at 9:15
    
@Asaf: Yes, I am. –  4math Mar 5 '11 at 9:19
    
Oh, any hints yet? I suppose this problem is somewhat complicated. –  4math Mar 5 '11 at 17:44
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1 Answer

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

  2. 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.

  3. 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.

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In (2), if $X,Y$ are finite then we get $\binom{Y}{X}$. –  Yuval Filmus Mar 6 '11 at 2:21
    
@Yuval Filmus: How do you prove that? –  4math Mar 6 '11 at 3:49
    
The images of the function form a subset of size $|X|$ of $Y$. Moreover, given the subset, the function itself is defined uniquely by the requirement that the function be order-preserving. –  Yuval Filmus Mar 6 '11 at 4:16
    
Between to finite sets of the same size, every constant function is order-preserving. Therefore, how is that there can only be one order-preserving function in that case? On the other hand, what would happen if in 2. we replace "order-preserving" for "strict order-preserving" or "order-reversing" –  4math Mar 6 '11 at 5:44
    
@4math, I interpret "order-preserving" to mean $x\leq y\iff \phi(x)\leq\pi(y)$, and I believe that this is what most people mean by that terminology (this is likely the same as what you mean by strict order-preserving). If you intend the property that $x\leq y\to \phi(x)\leq\phi(y)$, then you have merely a weak order homomorphism, and there will of course be more maps. –  JDH Mar 6 '11 at 19:32
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