MAT511 homework,          due Sept. 30, 2009

Let $A$, $B$, $C$ be sets.

1. Prove that $A\subseteq B$ if and only if $A - B = \emptyset$.

2. Prove that $C \subseteq (A\cap B)$ if and only if $C \subseteq A$ and $C \subseteq B$.

3. Prove that ${\mbox{${\mathcal{P}}\left(A \cap B\right)$}} = {\mbox{${\mathcal{P}}\left(A\right)$}} \cap {\mbox{${\mathcal{P}}\left(B\right)$}}$. You may use the results above.

4. Prove that ${\mbox{${\mathcal{P}}\left(A\right)$}} \cup {\mbox{${\mathcal{P}}\left(B\right)$}} \subseteq {\mbox{${\mathcal{P}}\left(A \cup B\right)$}}$.

5. Give an example where ${\mbox{${\mathcal{P}}\left(A\right)$}} \cup {\mbox{${\mathcal{P}}\left(B\right)$}} \ne {\mbox{${\mathcal{P}}\left(A \cup B\right)$}}$. What conditions are necessary on $A$ and $B$ to ensure that ${\mbox{${\mathcal{P}}\left(A\right)$}} \cup {\mbox{${\mathcal{P}}\left(B\right)$}} = {\mbox{${\mathcal{P}}\left(A \cup B\right)$}}$?

6. Show that there are no sets $A$ and $B$ for which ${\mbox{${\mathcal{P}}\left(A-B\right)$}} = {\mbox{${\mathcal{P}}\left(A\right)$}} - {\mbox{${\mathcal{P}}\left(B\right)$}}$.

7. Let $\mathcal{A}$ be the family of all sets of integers containing $10$. What are the sets ${\displaystyle \bigcup _{A \in \mathcal{A}} A}$ and ${\displaystyle \bigcap _{A \in \mathcal{A}} A}$ ? Justify your answer.

8. Let ${\displaystyle A_n = \left[ \frac{1}{n}, 2+\frac{1}{n} \right]}$. What are the sets ${\displaystyle \bigcup _{n \in ({\mathbb{N}}- \left\{{1,2}\right\})} \hspace{-1.2em}A_n}$ and ${\displaystyle \bigcap _{n \in ({\mathbb{N}}- \left\{{1,2}\right\})}\hspace{-1.2em} A_n }$ ? Justify your answer.

9. Let $\mathcal{A}$ and $\mathcal{B}$ be two pairwise disjoint families of sets. Let $\mathcal{C} = \mathcal{A} \cap \mathcal{B}$, and $\mathcal{D} = \mathcal{A} \cup \mathcal{B}$.
   1. Prove that $\mathcal{C}$ is a pairwise disjoint family of sets.
   2. Give an example where $\mathcal{D}$ is not a pairwise disjoint family of sets.
   3. Prove that if the sets ${\displaystyle \bigcup _{A \in \mathcal{A}} A}$ and ${\displaystyle \bigcup _{B \in \mathcal{B}} B}$ are disjoint, then $\mathcal{D}$ is a pairwise disjoint family.