# Math Help - Sets!

1. ## Sets!

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

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

Thanks

2. Given that $A^c \subseteq B$ then we know that
$U = A \cup A^c \subseteq A \cup B \subseteq U$ or $A \cup B = U$.

Given that $A \cup B = U$ then
$A^c \cap \left( {A \cup B} \right) = A^c \quad \Rightarrow \quad A^c \cap B = A^c \quad \Rightarrow \quad A^c \subseteq B.$

3. What about my use of the De Morgan LAws? IS that incorrect?

4. Originally Posted by shilz222
What about my use of the De Morgan LAws? IS that incorrect?
No, what you did is correct.