Then the right half says A-B so B takes a "bite" out of A. and A-C so C takes a "bite" out of A, and the leftovers from each of these are intersected together. Since the bite that B took will not be there, it has nothing to intersect, and since the bite that C took will not be there, it has nothing to intersect, so all that will remain from A-B intersecting A-C is A that was not removed in either operation. So A will have two "bites" out of it.
So I guess think of A as a cookie ^_^ and BUC as a person with two mouths taking a single bite, and (A-B) U (A-C) as two people who each take a bite out of the same cookie.
...Or you could probably search and find some mathematical equation to apply to this, or write a big long proof, or convert to set builder notation where you can manipulate easier.
As for the first one, I'm not familiar with the triangle in your notation, so I won't comment on it.