I was reading sets and came to some reserved letters for a few sets.
Two of them really confused me. They were –
$\mathbb N$ : For the set of natural numbers.
$\mathbb Z^+$ : For the set if all positive integers.
In my sense, both the sets contain $\{1,2,3,\dots\}$
Then, why are they considered different?
I searched a little on this topic and got this, but it doesn't tell anything about significance of two different sets.
Best Answer
You should be aware that some authors define $\mathbb{N}$ to include zero. This isn't of much consequence in itself since the properties of the set are preserved: there is a bijection between $\mathbb{N}$ with zero and $\mathbb{N}$ without zero, both are well-ordered, and so forth—effectively, we've done nothing but "relabel" the elements.
Only when we start adding structure to these elements does the distinction become important. For instance, if we define an addition $+: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, we might make $0$ an additive identity. Therefore, when one writes "$\mathbb{N}$" in such a scenario (most scenarios), then it should be made clear which definition is intended.
Now, if we take both to mean the set $\{1, 2, 3, \cdots\}$, then whether one writes $\mathbb{N}$ or $\mathbb{Z}^+$ is immaterial. However, using $\mathbb{Z}^+$ removes ambiguity since $\mathbb{Z}^+$ definitively does not include zero, and we would not have to go out of our way defining $\mathbb{N}$.