I'm needing a bit of guidance with the following question. I'm not entirely sure what the question is asking, so I'll put down what I know so far.

"Suppose (A, ≤ ) is a well-ordered set. LetB=R$\displaystyle ^A$ (sorry that should be superscript). Find an explicit linear order onB."

B is the set of functions which maps from A ->R. As we know A is well ordered, we can think of the map as {$\displaystyle a_1, a_2, ..$} ->R. So B is the set of functions {$\displaystyle f_1, f_2, ..$} : A ->R.

A andRare both linearly ordered by ≤, so by a definition we have the maps {$\displaystyle f_1, f_2, ..$}: (A,≤) -> (R,≤') are injective maps f: A ->Rwhich preserves the orders (a≤b => f(a) ≤' f(b)).

I now think I should prove that these injective maps satisfy reflexivity, antisymmetry and transitivity. According to the hint we're giving, the first two should be easy and the third trickier.

Please suggest how to do these proofs as it seems like whatever I write down is too simple. Also please tell me if what I've written so far is garbage!