Notation for union / intersection (in the same way $\pm$ stands for plus / minus) - is this a good idea? Note: $F$ is a class of sets.
I was solving a problem in Apostol's Calculus Volume 1. It is to show that
$$B-\bigcup_{A\in F} A=\bigcap_{A\in F}(B-A)\qquad\text{ and }\qquad B-\bigcap_{A\in F} A=\bigcup_{A\in F}(B-A)$$
so I thought that rather than repeating the problem just with $\bigcup$ and $\bigcap$ switched, why not create the notation of a circle with one half filled in to be like $\pm$ for the big cup and cap? In other words, see the attached image.  
The motivation for using this particular symbol should be clear because the one with a the bottom half colored black leaves only a semicircle at the top which is kinda $\bigcap$, and similarly the one with the top half colored black leaves a semicircular segment at the bottom which looks like $\bigcup$. But of course I do not mean to say that the one with the top colored black should denote $\bigcup$, nor that the one with the bottom colored black should denote $\bigcap$; rather, I am just using the symbols in the same way as say $\pm$ and $\mp$ are used in
$$\tan(a\pm b)=\frac{\tan a\pm \tan b}{1\mp \tan a\tan b},$$
to write two formulas in one. I hope this makes sense, please tell me what you think.
Or maybe there is already a convention for this, in which case I would like to hear about it.
Thanks!
 A: It's not awful, at least as an idea, but I would certainly say it's unnecessary, and I strongly dislike your particular notation choice. Basically it's a symbol that's too different and unintuitive to have to remember for too unimportant and uncommon a situation to be worthwhile.
In my experience, once one gets to mathematics classes, textbooks, etc. that assume a certain level of mathematical maturity, people would not write any statement like

$$\tan(a\pm b)=\frac{\tan a\pm \tan b}{1\mp \tan a\tan b}$$

but would write 

$$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$$
  (the same being true with the signs reversed, since $\tan$ is an odd function)

or maybe, if both versions were important to point out separately, 

$$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}\qquad\textsf{and}\qquad\tan(a-b)=\frac{\tan a-\tan b}{1+\tan a\tan b}$$

or just not comment on the "flipped" version at all:

$$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$$

The same is true with $\bigcap$ and $\bigcup$, or dualizing basic things in category theory, or anything similar.
A: That's a very creative idea. I don't know of an established set theory symbol for this (e.g. http://www.rapidtables.com/math/symbols/Set_Symbols.htm doesn't even define such an "operation") and it doesn't seem to be a standard symbol in $\LaTeX$. One problem I can see is that the symbol $\ominus$ is quite widely used to denote the symmetric difference of sets, which is certainly not the same as what you are trying to symbolise. 
