Example Question #1 : Well Orderings And Transfinite Induction
Determine if the following statement is true or false:
Let and be well ordered and order isomorphic sets. If and are also order isomorphic sets, then and are also order isomorphic.
This is a theorem for well ordered sets and the proof is as follows.
First identify what is given in the statement.
1. The sets are onto order isomorphisms
2. The goal is to make an onto order isomorphism. Let us call it .
Thus, can be defined as the function,
To show is a well defined, one-to-one, function on since and are one-to-one, the following is performed.
Thus, is one-to-one.
Now to prove in onto if
proving is onto .
Lastly prove ordering.
Thus proving is isomorphic. Therefore the statement is true.