Many of the 16 binary set operations (corresponding to the 16 logical connectives) don’t seem to have common dedicated symbols. Instead, they are often simply expressed in terms of $\cup, \cap, \setminus$, etc.
My question is whether there is any known precedent of a symbol for the operation $A \cup B^\complement$. This would complete the following (where ? is a placeholder for the operation in question):
- $A \setminus B = A \cap B^\complement$
- $A \operatorname{?} B = A \cup B^\complement$
For more context (based on discussion in the comments below):
In everyday work, of course it makes sense to use just a few symbols (e.g. complement, $\cup, \cap, \setminus$) and express other operations in terms of these. This approach represents a practical tradeoff between two extremes:
- At one extreme, only one symbol is needed (just NAND or NOR, cf. functional completeness). The drawback is that most expressions will be quite large when written this way.
- At the other extreme, we would have a unique symbol for each of the 16 operations. Expressions may be written more compactly with these symbols, but then there is more to memorize.
(Why are the complement, $\cup, \cap, \setminus$ operations in particular given special status? My only guess is that they correspond with concepts in spoken language: NOT, OR, AND, WITHOUT, which, in turn, may correspond to which operations are most naturally computed by our brains, but I digress...)
A comment below suggests simply using $(B \setminus A)^\complement$ to represent $A \cup B^\complement$, but the same could be said of any operation, e.g. why do we need a symbol for set difference $A \setminus B$ when we can just write $A \cap B^\complement$? Or why do we need a special logical connective for implication (which actually corresponds to the set operation in question here)?
My point is that sometimes things can be written more concisely with dedicated symbols for the uncommon cases. It may be rare that these symbols are needed, but in those instances where one of the uncommon set operations is heavily utilized ($A \cup B^\complement$ in my case here), it would be nice to have some agreed-upon symbol. And here I simply want to know if there is any precedent in the literature for those weird cases.
The diagram below (originally found on Wikipedia here or here) shows that all 16 logical connectives have dedicated symbols. Why not extend this luxury to the corresponding set operations?