Sets!

**shilz222** · August 14th 2007, 10:39 AM

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

**Plato** · August 14th 2007, 11:00 AM

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

**shilz222** · August 14th 2007, 11:04 AM

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

**Plato** · August 14th 2007, 11:09 AM

Originally Posted by shilz222

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

No, what you did is correct.

Thread: Sets!

LinkBack

Thread Tools

Display

Sets!

Similar Math Help Forum Discussions

Open sets and sets of interior points

Metric spaces, open sets, and closed sets

3 sets of 5 pieces of data. Compare and contrast the three sets. Comment briefly on t

Approximation of borel sets from the top with closed sets.

how to show these sets are 95% confidence sets

Search Tags