general topology - Order type between two sets and bijection?

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?