I'm sorta lost and sorta have an idea, so I don't know what to do really. The problem is to show that the closure operation has the following properties:
(a) if E1 is a subset of E2, then closure(E1) is a subset of closure(E2)
(b) closure(E1 U E2) = closure(E1) U closure (E2)
(c) closure(E1 intersect E2) is a subset of closure(E1) intersect closure(E2)
I do not need the full out proofs, but I do need some direction...
on (a) I'm not sure how to show it at all, it's the one I'm totally lost on.
(b) I get part of this one. I know you start with an element that is in the left side which leads to it being in E1 or E2 or (E1 U E2)', or the accumulations points of the union of E1 and E2 but do not know how to rewrite it so I can get the element in E1' and E2', or if (E1UE2)' is just E1' U E2.
(c) I can get this once I can get b.