Are $\mathbb{N}$ is isomorphic to $\mathbb{Q}$?

Question

Are $\mathbb{N}$ isomorphic to $\mathbb{Q}$?

There are any difference between isomorphism and cardinal equality? If $X$ and $Y$ are two sets and $\text{Cardinal}(X)=\text{Cardinal}(Y)$, is $X$ isomorpic to $ Y$?

Isomorphic as what? In the category of sets, isomorphisms are precisely the same as bijective maps. — Tobias Kildetoft, Jul 03 '15 at 09:18

drhab · Answer 1 · 2015-07-03T09:32:18.270

3

If you are looking at $\mathbb N$ and $\mathbb Q$ purely as sets then the answer is 'yes'. In that case bijections are isomorphisms. If structure is added (e.g. order or composition) then more is needed. The bijections that preserve and reflect the structure are isomorphisms.

edited Jul 03 '15 at 09:32

answered Jul 03 '15 at 09:25

drhab

151,093

In first case, we can looking a set, such as an $(0)-type$ structure? (Ring is $(2,2,0)$ types, group is $(2,0)$, Boolean algebra is $(2,2,1,0,0)$) – Jamal Farokhi Jul 03 '15 at 09:38
Actually we are looking at categories. – drhab Jul 03 '15 at 13:12

Are $\mathbb{N}$ is isomorphic to $\mathbb{Q}$?

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