The unambiguous notations are: for the positive-real numbers
$$
\mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\} \,,
$$
and for the non-negative-real numbers
$$
\mathbb{R}_{\geq 0} = \left\{ x \in \mathbb{R} \mid x \geq 0 \right\} \,.
$$
Notations such as $\mathbb{R}_{+}$ or $\mathbb{R}^{+}$ are non-standard and should be avoided, becuase it is not clear whether zero is included. Furthermore, the subscripted version has the advantage, that $n$-dimensional spaces can be properly expressed. For example, $\mathbb{R}_{>0}^{3}$ denotes the positive-real three-space, which would read $\mathbb{R}^{+,3}$ in non-standard notation.
Addendum:
In Algebra one may come across the symbol $\mathbb{R}^\ast$, which refers to the multiplicative units of the field $\big( \mathbb{R}, +, \cdot \big)$. Since all real numbers except $0$ are multiplicative units, we have
$$
\mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,.
$$
But caution! The positive-real numbers can also form a field, $\big( \mathbb{R}_{>0}, \cdot, \star \big)$, with the operation $x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) }$ for all $x,y \in \mathbb{R}_{>0}$. Here, all positive-real numbers except $1$ are the "multiplicative" units, and thus
$$
\mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,.
$$
Both symbols $\setminus$ \setminus and $-$ - are used for denoting set difference: $$A\setminus B = A - B = \{ x \mid x \in A,\,x \not\in B \}.$$
I, particularly, prefer $A \setminus B$. In some contexts, we can have something like: $$A-B = \{ x-y \mid x \in A,\, y \in B \},$$ so sticking to $\setminus$ there is zero chance of confusion.
Best Answer
In an elementary context, $\Bbb W$ means the set of whole numbers. Some books have it as
$$\Bbb W=\{0,1,2,\ldots\}$$
while others have it as
$$\Bbb W=\{1,2,\ldots\}$$
Because of the ambiguity, I recommend that you avoid the use of $\Bbb W$. For the second meaning use $\Bbb Z^+$. There still is no perfect abbreviation for the first. Either meaning is also called the Natural numbers, although usually the Whole numbers are meant to be different from the Natural numbers. Again, we see the ambiguity: even if the Whole and Natural numbers are different, which is which? See the Wikipedia articles for a variety of notations for these sets, most of which are far from perfect.
I have occasionally seen the phrase "whole numbers" used for the Integers, which includes negative numbers such as $-1,-2,\ldots$ and is usually written $\Bbb Z$. But I have never seen the notation $\Bbb W$ used in that way.
As @RenatoFaraone points out, in an advanced context $W(x)$ probably means the Lambert W function. But I have never seen that written in the "blackboard bold" style that $\Bbb W$ uses.