Correct mathematical notation

Question

In mathematical notation, if I define a set of variables as $X$ (in bold), should I name subsets $X_A$ and $X_B$ also in bold or not?

Which is the best form of following a correct notation?

Answer 1

To start off, let me say what you already know: ultimately notation doesn't follow strict rules. Good notation helps the audience read text naturally by giving them hints about what to expect. If you always write capital letters for matrices and lowercase letters for vectors, then the expression $Ax = b$ is instantly clear, and $aX = B$ less so.

In this case, usually subsets of a set are "on the same level" as the original set: that is, they are collections of the same type of object. Hence, it makes sense to use similar typographical conventions for a set and its subsets. If a set is called $X$, you might expect an element to be called $x$; then a subset might be $X'$ or $X_A$, and it still feels natural if $x \in X'$ or $x \in X_A$.

Exceptions exist, in particular when there are special names involved. Don't use blackboard bold characters for subsets of $\mathbb R$, for example.