1. Define $V_0=\emptyset$, $V_1={\mathcal P}(V_0)$, $V_2={\mathcal P}(V_1)$, and $V_3={\mathcal P}(V_2)$.
(a) Express the sets $V_0, V_1, V_2$ and $V_3$ in roster notation.
(b) Draw a directed graph whose ``dots'' (or ``vertices'')) are the sets in $V_3$ and whose directed edges, written $x\to y$, are interpreted to mean $x\in y$. (Hint: your graph should have four ``dots'' and four edges.)

2. Find sets $A$ and $B$ satisfying the given conditions.
(a) $A\in B$ and $A\not\subseteq B$.
(b) $A\in B$ and $A\subseteq B$.
(c) $A\notin B$ and $A\subseteq B$.

3. Show that $\bigcup {\mathcal P}(x) = x$. (Remember the proper way to show two sets are equal!)

Solution sketches (dischw1sol.tex),
Solution sketches (dischw1sol.pdf),

1. We have explained why the Russell class ${\mathcal R}=\{x\;|\;x\notin x\}$ is not a set. Explain why the following classes are also not sets.
(a) The class $\mathcal C$ of all sets.
(b) The class $\mathcal D$ of all $1$-element sets.
(Hints: For (a), show that the assumption that $\mathcal C$ is a set allows you to construct $\mathcal R$ as a set. For (b), show that the assumption that $\mathcal D$ is a set allows you to construct $\mathcal C$ as a set.)

2. Your friend offers a wager that, under the Kuratowski encoding, the ordered pair $(0,1)$ equals the natural number three. Should you take the wager? Explain.
(More detail: You friend is offering money to support the claim that the set $(0,1)$ equals the set $3$. If you believe that your friend is correct, then you should not take the wager. Otherwise, to take the wager, you should offer money saying that this is false. Whoever is correct will get all the money.)

3. Show that $\emptyset\times A = \emptyset$.