Is $\mathbb{N} = \mathbb{Z}^+$?

Question

Because $\{1,2,3,4,\ldots\}$ contains all natural numbers, which are also all positive integers.

Answer 1

In some reference it is true,or $\Bbb{Z}^+=\Bbb{N}\cup\{0\}$

Answer 2

Depends. In a lot of modern mathematics, $\mathbb N=\{0,1,2,\dots\}$. In particular, since you've tagged the question "set theory," set theorists almost always include $0$ in $\mathbb N$.

If $\mathbb N$ does not include $0$, you are correct.

Often, mathematicians choose to write $\mathbb Z^+$ simply to avoid ambiguity.

There is an infuriating practice in some linear programming to include $0$ in $\mathbb Z^+$, and using $\mathbb Z^{++}$ to mean the "really positive integers."

In other words, it depends on your definitions. $\mathbb N$ and $\mathbb Z^+$ are both just notation. What they mean varies from source to source.

There's one last pedantic wrinkle, actually. Usually, the integers are defined in terms of ordered pairs of natural numbers. In that sense, it is not even true that $\mathbb N\subset \mathbb Z$. Rather, there is an embedding of $\mathbb N$ into $\mathbb Z$. We usually skip this sort of pedantry, and when we are talking about integers, we treat the symbol $\mathbb N$ to be the image of this embedding of the natural numbers into the integers.

Answer 3

Symbols have meaning because we agree on what the symbols mean, or at least we say what we're using the symbol to mean so that other people understand even if we use symbols in ways contrary to how they would use them. That's why you write things like "if $n$ is an integer" or "let $x$ be a real number." You don't just assume people will understand $n$ is an integer and $x$ is a real number, standard as these assignments may seem.

If you really want to use $\mathbb{N}$ to mean only the positive integers, then you should say so, because some people might think that you're including 0. I could use an arbitrary symbol like $\spadesuit$ to represent all the integers greater than 0, but I would probably be quite alone in this. That would still be better than using $\mathbb{N}$ and assuming people will just know that I mean only the positive integers.

But even with terminology there are problems, since apparently for some people there is a distinction between "positive" and "strictly positive" numbers. So for you it might be best to write something like "$\mathbb{N} = \{1, 2, 3, 4, \ldots\}$"

P.S. For a little comic relief, Google Selina Meyer clarifies recent mis-speaking.