Suppose we have two well ordered setsandand a mappingthat maps elements ofto elements of

f is order isomorphic if f is a bijection (one to one and onto) and f satisfies

Ifis order isomorphic then so is

Well ordered setsandwith the same number of elements are automatically order isomorphic because both sets can be arranged in order and f defined as

The sets of odd natural numbersand even natural numbersare order isomorphic, since both sets can be arranged in increasing order andis a one to one and onto mapping from O to E.

The set of natural numbersand the set of natural numbers rearranged asare not order isomorphic since in order forto be onto we must havesomapsonto the odd natural numbers andis not onto.

We can extended order isomorphisms in the natural way to any number of well ordered sets and order isomorphisms between them.