Other than $\setminus$ and $-$, are there any other notations for the set-theoretic difference of sets?

Question

Let $X$ denote a set, and suppose that $B$ and $A$ are subsets thereof. Then the set-theoretic difference of $B$ and $A$ may be denoted in any of the following ways:

$$B \setminus A, \qquad B - A, \qquad B \cap A^c$$

I'm not entirely happy with any of these options, however:

1. $B \setminus A$ "looks" contravariant in the first argument and covariant in the second, but in actuality the opposite is true. Furthermore, if $A$ and $B$ are subsets of a non-commutative monoid $M$, I like to write $B \setminus A$ for the set of all $c \in M$ such that $Bc \subseteq A$. Notice that this is contravariant in the first argument and covariant in the second.

2. $B - A$ is a good option unless $X$ has an Abelian group structure, in which case there's a potential notational conflict with $\{b-a \mid b \in B, a \in A\}$.

3. $B \cap A^c$ is a good way of proving things about set-differences algebraically, but I don't think its a good idea to get rid of a symbol for set-theoretic difference altogether. It would be like getting rid of "$p$ if $q$" in favor of "$p$ or not $q$." Although this algebraic reduction can be useful, I think it kind of undermines clarity and readability.

I'm currently using $\setminus$, but I'd like to replace it with something else. The notation $A / B$ isn't a good option, because it looks like a quotient of the structure $A$ by the subobject $B$.

Question. Other than $\setminus$ and $-$, are there any other notations for the set-theoretic difference of sets?

Answer 1

There's a notation that I've seen in a point-set topology book: $\mathrm{C}_S \,(\mathrm{T})$ is notation for the complement of $\mathrm{T}$ relative to $\mathrm{S}$, or $\mathrm{S} \setminus \mathrm{T}$. See https://proofwiki.org/wiki/Definition:Relative_Complement.

Answer 2

A standard notation in Boolean calculus is to use $\bar A$ for the complement of $A$, $AB$ for $A \cap B$ and $A + B$ for $A \cup B$ and hence $B\bar A$ for $B \setminus A$ .

That being said, the most appropriate notation depends on the context. If you are working on Hausdorff's nested difference hierarchy, the notation $B - A$ proves to be very convenient (together with $B + A$ for the union). Actually, I tend to use more and more $B - A$ instead of $B \setminus A$. If $X$ has a group structure, there is ambiguity only if you use the group operation and the set difference at the same time. In this event, you may wish to use some temporary notation like $\dot{-}$ with a dot on top.