Note: F is a class of sets.
I was solving a problem in Apostol's Calculus Volume 1. It is to show that
B−⋃A∈FA=⋂A∈F(B−A) and B−⋂A∈FA=⋃A∈F(B−A)
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 ⋂, and similarly the one with the top half colored black leaves a semicircular segment at the bottom which looks like ⋃. But of course I do not mean to say that the one with the top colored black should denote ⋃, nor that the one with the bottom colored black should denote ⋂; rather, I am just using the symbols in the same way as say ± and ∓ are used in
tan(a±b)=tana±tanb1∓tanatanb,