Thanks for your effort and patience, sir (or mam)! I think I've got it now, I'll just do some more examples to get it drilled into my head.
i think you want to get away from saying things like: "everything shaded but B".
the reason is, that usually, A and B are part of some larger collection, so "everything but B" isn't the same as "A", or even "A-B".
you wind up with two sets:
A∩B, and A∩B'.
note that both of these sets have part of A in them (well, hopefully. it could happen that B is so far away from A that there is no overlap, or that A is entirely inside B, so that there is no overlap between A and "not-B").
so one set is the part of A that overlaps with B, and the other set is the part of A that overlaps with everything else BUT B. put these two parts of A together, and you get A.
there is a distributive law of U and ∩ that applies here:
(A∩B)U(A∩B') = A∩(BUB').
but BUB' is "everything", sometimes called the "universe of discourse", or the "containing set" (in a Venn diagram, the rectangle containing the overlapping circles).
and the intersection of A with "everything", is just A.