What is a proof of set theory problems?

[congtongsat]

Problem:

(i)A\subseteqB \Leftrightarrow A\cupB = B
(ii) A\subseteqB \Leftrightarrow A\capB = A

and

For subsets of a universal set U prove that B\subseteqA^{c} \Leftrightarrow A\capB = empty set. By taking complements deduce that A^{c}\subseteqB \Leftrightarrow A\cupB = U. Deduce that B = A^{c} \Leftrightarrow A\capB = empty set and A\cupB = U.

Can't wrap my head around the last question at all. The i and ii seem simple but I'm just not getting it to work.

 

[clamtrox]

(i) If A is empty, claim is true trivially. If it's not, then take an element of A, x \in A.

Suppose A is a subset of B. What does this mean for x? Use the definition of the cup operation :) Then suppose A \cup B = B and do the same.

For (ii) you might want to assume that A is not empty because the thing you're trying to prove does not generally hold if both A and B are empty (mathematicians are weird...)

For the last bit, use again some element of b, y \in B. Show that if y \in A^c then it can't be in A.