In both$\mathbb Z$ and $\mathbb R$
1) if $a,b\in S, a

Question

In both$\mathbb Z$ and $\mathbb R$
1) if $a,b\in S, a< b,~~then~~a\neq b$
2)$if ~~a<b~~and~~b<c,~~then~~a<c$
Do they isomorphic ? (in Z and in R) 
Please, help.

loser66 · Answer

I feel like they are not one to one but don't know how. I establish 
$f:\mathbb Z\rightarrow \mathbb R$ , $f: (a,b)\mapsto a/b$ I can see that f is not one to one. But how to put this logic into the problem?

loser66 · Answer

On the other hand, if the relation is "less than", then, it works well for both set, hence they are isomorphic. Ha!!!!! I confused myself.

ganeshie8 · Answer

it must be one-to-one too right
and integers and reals have different cardinalities 
idk what that means for present problem..

loser66 · Answer

Modern geometry is my course. This is the very first section on the book and I got stuck everywhere. hihihi.....