Prove that $\displaystyle A^{c} \subseteq B \Leftrightarrow A \cup B = U $ (universal set). From a previous part of the problem I already proved that $\displaystyle B \subseteq A^{c} \Leftrightarrow A \cap B = \emptyset $. The problem then says to take complements to deduce the former result.

So I said that $\displaystyle A^{c} = B $ and so $\displaystyle A \cup B = U $ by definition. In the other direction I am given that $\displaystyle A \cup B = U $. I tried using DeMorgans Laws: $\displaystyle (A \cup B)^{c} = A^{c} \cap B^{c} = \emptyset $ and so $\displaystyle A^{c} \subseteq B $? Is this correct? Did I do what the hint told me?

Thanks