Set question - $ ℤ^+ = ℕ$

Question

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$

if not, why?

Thank you in advance! I appreciate your help!

Answer 1

You need to know the definitions.

$\mathbf{N}$ usually includes zero and $\mathbf{Z}^+$ usually does not. But occasionally people define $\mathbf{N}$ to exclude zero or $\mathbf{Z}^+$ to include zero, or both.

I often use $\mathbf{Z}^>$ (respectively $\mathbf{Z}^{\geq}$) for excluding (resp. including) zero, as I've never seen any ambiguity with those.

(or sometimes the longer forms $\mathbf{Z}^{>0}$ and $\mathbf{Z}^{\geq 0})$

Answer 2

If $\mathbb{N} \!\,$:= {$1,2,3,4,5,...$} and ℤ+ ={$1,2,3,4,5,...$}, then we can show that $\mathbb{N} \!\,$ is isomorphic to ℤ+.

So we can think of them as "equal".

Obviously, if you define $\mathbb{N} \!\,$ or ℤ+ otherwise then this won't hold.

Usually, $\mathbb{N} \!\,$ is defined not to include $0$ as originally stated.

Answer 3

It is a matter of convention. Always keep in mind that "positive" $\neq $ "non-negative". I like to consider $0 \in \mathbb{N}$, because of the notation $\mathbb{N}^*$, which means $\mathbb{N} \setminus \{ 0 \}$. And indeed $\mathbb{Z}^+$ and $\mathbb{N}^*$ are isomorphic.