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\homework{6: Set Theory}{Due Friday, Week 3}

Do \textbf{one} of the \textbf{three} problems below!

\begin{enumerate}
\item A \textbf{singleton} is any set of the form $\{x \}$: that is, any set that contains exactly one element.  Suppose that we could form the set
\begin{align*}
S = \{ A ~|~ A \textrm{ is a singleton set.}\}.
\end{align*}
Show that we can use this claimed set to create a contradiction\footnote{This is a little similar to how in class we looked at Russell's paradox, linked \href{http://en.wikipedia.org/wiki/Russell's_paradox}{here}, which pointed out how the object $R = \{ A ~|~ A \notin A\}$ creates a contradiction if it is a set.}.
\item The \textbf{generalized union axiom} is the following axiom:
\begin{axiom}
(Axiom of Union, full version.)  Given any set $A$, there is a set $C$ whose elements are exactly the members of the members of $A$.  In symbols:
\begin{align*}
\forall A ~ \exists C ~ [(x \in C) \Leftrightarrow (\exists A'(A' \in A) \wedge (x \in A'))].
\end{align*}

We denote this set as
\begin{align*}
\bigcup_{A' \in A} A'.
\end{align*}
\end{axiom}
In class, we gave instead the ``weakened'' version of this axiom:
\begin{axiom}
(Axiom of Union, simple version.)  Given any two sets $A,B $, we can make a set whose members are those sets belonging to either $A$ or $B$ (or both!)  In symbols:
\begin{align*}
\forall A, B ~ \exists C ~ \forall x [x \in C \Leftrightarrow ((x \in A) \wedge(x \in B)].
\end{align*}
We write this set $C$ as $A \cup B$, for shorthand.
\end{axiom}

Using literally only the axioms we've given in class other than the simple union axiom, show that the stronger union axiom can ``prove'' our weaker union axiom.  (In other words: start with the axioms we've gone through in class - (simple union) + (generalized union).  Use just these results to prove the simple axiom of union.)
\item \begin{enumerate}
\item Find two sets $a, B$ such that $a \in B$, but $\mathcal{P}(a) \notin \mathcal{P}(B)$.
\item Prove that if $a \in B$, we must have 
\begin{align*}
\mathcal{P}(a) \in \mathcal{P}\left(\mathcal{P}\left( \bigcup_{b \in B} b \right) \right).
\end{align*}
\end{enumerate}
\end{enumerate}
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