I want to show that $$ \{1,2\}\times Z_+\ \text{and} \ Z_+ \times\{1,2\}\ \text{have different order type}$$
If we define $$f(i,j)=(j,i)\ \text{for}\ i\ \text{in }\{1,2\}\ \text{and} \ j\ \text{in}\ Z_+$$
It seems like that this is bijective map between two sets.
However, to show that they are not order isomorphic, how shall I start to show that bijection does not preserve ordering?
It seems like that the way I defined the bijection is not the only way.
I am wondering if there exists any bijection between two sets and that bijection does not preserve order, can I conclude that they have different order type?
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