Here is a problem I am doing. Suppose and let
Prove that and therefore .
Velleman gives some hints at the back of the book. To prove ,
he suggets a function defined as
He suggets some things to prove that h is one to one. I could do that. Now to prove
that h is onto, he says , suppose R is a total order on A. Define
by the formula .
He asks the reader to show that
I could show this. Then he says use this fact to show that
and . I could show that g is one to one. To prove that
, I should also show that g is onto. But I have not been able
to show this. Input is welcome.
I have to first establish that i.e. g is one to one and onto function
from to . You are right that on finite sets of equal cardinality, every one-to-one function is onto. I think I said at the beginning that
Since I have proven that g is one to one, I can use the above fact to establish that
g is onto. Correct ?