Topology and order type problem

[g1990]

Homework Statement

Both {1,2}x Z+ and Z+ x {1,2} are well-ordered in the dictionary order. Are they of the same order type? Why or why not?

Homework Equations

To be of the same order type, we must be able to construct a bijection that preserves order, that is, x<y => f(x)<f(y). well-ordered means that every nonempty subset has a minimal element.

The Attempt at a Solution

I know I need to find a bijection between the two, but I can't seem to think of one. I can get a bijection from Z+ x {1,2} to Z+, but I can't find one from the other one to Z+

 

[g1990]

oh yeah- Z+ is the set of positive integers

 

[Hurkyl]

I know I need to find a bijection between the two,

Are you sure?

 

[g1990]

well, I don't know how I can prove that there is no bijection. Again, I know I can find a bijection from Z+ to {1,2}xZ+, but listing the other set in order would entail TWO infinities. It would be (1,1),(1,2),(1,3)... (2,1),(2,2),(2,3)... is that enough to say that Z+ x {1,2} cannot be bijected into Z+ and therefore there is no bijection between them?

 

[Hurkyl]

That's the basic idea.

However, given that you're just learning this stuff, your teacher probably expects you to both provide the basic idea and fill in the details to turn it into a proof.

e.g. why would having "two infinities" make a difference? Are you sure Z+ doesn't have "two infinities"?

 

[g1990]

okay- thanks!

 

[Hurkyl]

Oh, and for the record -- you want the term "order isomorphism", not "bijection". While every order isomorphism is a bijection, not all bijections are order isomorphisms.