Is $\mathbb{N} = \mathbb{Z}^+$? Because $\{1,2,3,4,\ldots\}$ contains all natural numbers, which are also all positive integers.
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.
In some reference it is true,or $\Bbb{Z}^+=\Bbb{N}\cup\{0\}$