$\mathbb {R,Z}$ etc. are imitating the way we write bold R, Z on a blackboard (hence the name, blackboard bold). It can be argued that when TeXing (not actually writing on a blackboard), you should write $\mathbf {R,Z}$ instead (since that's what $\mathbb {R,Z}$ are meant to represent on a blackboard, in the first place!), and I, for one, do just that most of the time.
They are written in bold to make the name distinct, because $R,Z$ may be used to represent other, more locally defined objects, while bold letters are rarely used as local variables. As to why are the particular letters are used, the $\bf R$ is probably self-explanatory, while $\bf Z$ originates from German (Zahlen).
$\bf K$ as a dummy field name also comes from German (Körper), and in this case bold is likely used to imitate $\bf R,C$ and to indicate that it is "the" background field when it is fixed in the context, so it is, at least locally, as fundamental as $\bf R,C$ are (e.g. in linear algebra and algebraic geometry). It is less often used in that way when we consider many distinct fields ard rings, like in abstract algebra (where letters starting with $K$, and continuing with $L$, and sometimes $M,N$, are still often used to denote fields, but are rarely bolded).
The triangle is almost certainly being used to for symmetric difference:
$$A\mathrel{\triangle}B=(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus(A\cap B)\;.$$
The $P$ might be the power set operator, in which case the whole thing is the set of all subsets of $A\mathrel{\triangle}B$:
$$P(A\mathrel{\triangle}B)=\{X:X\subseteq A\mathrel{\triangle}B\}.$$
However, a script $P$ is more often used for this purpose, and as has just been pointed out to me, you used the probability tag, so $P(A\mathrel{\triangle}B)$ is probably the probability assigned to the subset $A\mathrel{\triangle}B$ of your sample space.
Best Answer
Given any monoid (that is a set $A$ equipped with an associative operation $\cdot$ and an identity $1$), we can define "finite products" roughly by:
$$\prod_{i=1}^n a_i = a_1\cdot a_2\cdot \dots \cdot a_n$$
where $\prod_{i=1}^0 = 1$.
Possible monoids are for example $(\mathbb{R},\cdot, 1)$ yielding "$\prod$", $(\mathbb{R},+,0)$ yielding "$\sum$" or $(P(S), \cup, \emptyset)$ yielding "$\bigcup$" and so on and so forth.
So, we can also view a monoid as a set $A$ together with a map $A^* \to A, (a_n) \mapsto \prod_{i=1}^n a_i$ taking lists (words, tuples) of elements of $A$ to elements of $A$.
Occassionally however, we may find "maps" (broadly speaking) which not only accept finite lists, but also infinite lists or even bigger families of elements as objects.
For example, a complete lattice is a set $A$ equipped with maps $\bigvee$ and $\bigwedge$ taking abitrary families of elements of $A$ to elements of $A$.
Intuitively, if you take the set of all "small sets" (this is usually realized as a proper class) as the set $A$, then you get a complete lattice with operations $\bigcup$ and $\bigcap$ called union and intersection, which take families of elements of $A$ (that is sets of sets) to sets.