Difference between $\mathbb{Z}^+$ and $\mathbb{N}^+$

Question

I was reading through Mathematical Foundations of computing (preliminary course note by keith Schwarz pg 15 ) and noticed there was a definition for sets of positive natural numbers $\mathbb{N}^+ = \{ 1,2,3...\}$, noting that $0 \notin \mathbb{N} ^+ $ and I also read in Mathematics for computer science ( by Eric Lehma pg 14) that $\mathbb{Z}^+$ represents positive integers having the same set $ \{1,2,3,\dotsc\}$. So my question is, are they the same set? Is there any difference between the set of positive natural numbers and positive integers?

Answer 1

Yes, the two sets are equivalent.

The natural numbers and the integers are constructed by very different methods, so by some arguments they are not even the same mathematical object. (For instance, one might argue that integers are the quotient set of pairs of natural numbers under an equivalence relation!)

However, at the end of the section of those constructions, authors traditionally note (or prove) that all such constructions of number systems are isomorphic to each other, and so they define completely abstract sets $\mathbb N$, $\mathbb Z$, $\mathbb Q$,and $\mathbb R$ such that we can legitimately say that $\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R$. In that sense (which is the typical sense), it would be completely legitimate to say that $\mathbb N^+=\mathbb Z^+$.