Topology and order type problem

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Homework Help Overview

The discussion revolves around the comparison of the order types of the sets {1,2} x Z+ and Z+ x {1,2} in the context of well-ordering and bijections. Participants explore whether these two sets can be considered of the same order type and the implications of their structures.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need for a bijection that preserves order between the two sets. There is uncertainty about how to construct such a bijection and whether it is possible at all. Some participants question the implications of having "two infinities" in the context of these sets.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on the existence of a bijection and the concept of order isomorphism. Some guidance has been provided regarding the terminology and the need for a more detailed proof, but no consensus has been reached on the main question.

Contextual Notes

Participants are navigating the complexities of order types and well-ordering, with some expressing confusion about the definitions and implications of their findings. The distinction between bijections and order isomorphisms is also under consideration.

g1990
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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+
 
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oh yeah- Z+ is the set of positive integers
 


g1990 said:
I know I need to find a bijection between the two,
Are you sure?
 


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?
 


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"?
 


okay- thanks!
 


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.
 

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