Question

Prove or Disprove: −N := {z ∈Z : −z ∈N} has the Upper Well Ordering Property.

Answer 1

A set of numbers has the Upper Well Ordering Property if and only if every non empty subset has a greatest element.

Answer 2

i know that N has a least element

Answer 3

i think is wrong because N is infinite so doesn't have a greatest element. is that right?

Answer 4

I would say yes because you are dealing with the negative numbers not the naturals. Clearly all negative numbers are bouded above by 0 and the situation is analogous to natural numbers bounded below by 0 (or 1 depending on definition).

Answer 5

i see your point!

Answer 6

It does have the upper well ordering property. Let \( A \subset -N\). Then consider \(-A=\{-a:a \in A\} \). By the (lower?) well ordering property of N, -A has a least element, and so the negative of that least element is the greatest element of A. As A is arbitary, any subset of -N has a greatest element and so -N has the upper well ordering property

Answer 7

i'm a little confuse!

Answer 8

it has upper well ordering property or not? because you said in the beginning no and in the end yes!

Answer 9

Umm, i said it does have the upper well ordering property in the beginning and at the end.

Answer 10

thanks