The set of all prime numbers is usually denoted by $\mathbb{P}$. The set of all composite numbers, however is not denoted by $\mathbb{C}$, given the ambiguity with the set of complex numbers.
What is the correct (usual) way of denoting the set of composite numbers (with a single symbol)?
EDIT – an example:
Given a function $f$ that has a "prime version" and a "composite version", one may denote the "prime $f$" function by $f_{\mathbb{P}}$, but the "composite $f$" function cannot be denoted by $f_{\mathbb{C}}$ since it creates ambiguity with the set of complex numbers. What symbol would one use in this case to denote the "composite $f$" function?
Best Answer
The hard truth, as the comments show, is that there is no standard choice of a single symbol for denoting the composite numbers. There aren't very many single symbols available, so any time a person adopts a new single symbol to stand for something, there is a huge risk of ambiguity.
There are many ways of dealing with this risk in mathematical writing.
One is that mathematicians, as a whole, are not easily won over to new notations with one symbol when a simple notation of 3 symbols might do. For example, if I grant you (which I cast doubt on in my comment) that $\mathbb{P} \subset \mathbb{N}$ is a good and standard notation for the prime numbers as a subset of the natural numbers, then the notation $\mathbb{N}-\mathbb{P}$ or $\mathbb{N} \setminus \mathbb{P}$ for the composite numbers is a perfectly clear, it communicates instantly what is meant, and it is only two symbols more than being just one symbol. What's not to like about that?
Well, maybe you are writing a paper in which you need a symbol for the composite numbers, and you need to use it a lot, and you don't want to clutter up the paper with three symbols over and over and over when one symbol can be chosen (I myself find this unconvincing, because 3-to-1 is such a tiny ratio, but then I imagine a 42-to-1 situation and I am more convinced). Or, as your more recent edit suggests, what you really want is to use the symbol as a subscript (this is more convincing for a 3 symbol notation, although I've seen subscripts like that sometimes). In this situation, you are free to pick your own 1 letter symbol to stand for the composite numbers, even if it clashes with standard notation. But there's a catch.
For example, everybody loves $x$. There are even jokes about $x$. And there is a significant proportion of mathematical papers, or proofs within papers, in which the 1-letter symbol $x$ is used to stand for something special. Often $x$ stands for a real number, but just as often it stands for something else. So there is a gigantic risk of ambiguity when one uses the symbol $x$. The way we deal with this in our writing is that we state, clearly and carefully to avoid all possible ambiguity, what $x$ stands for in the limited context of our own paper or our own proof.
So, in whatever you might be writing, you should feel free to state, carefully, clearly, what $\mathbb{C}$ stands for in the limited context of what you are writing. If what you are writing uses both the composite numbers and the complex numbers, well then you have an ambiguity problem to solve, and perhaps you really do not want to use $\mathbb{C}$ for the composite numbers because yes, it is standard for the complex numbers.
But if what you are writing has nothing at all about complex numbers, well then, you should feel free to use $\mathbb{C}$ for the composite numbers, as long as you state clearly and carefully and without ambiguity what $\mathbb{C}$ stands for in your paper. And if somebody screams at you for abusing $\mathbb{C}$, you may want to pick something else. Just be clear and careful about it.