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\textbf{Name:} \smallskip  \\
\textbf{Collaborators:} \smallskip \\ %%% List anyone with whom you discussed the problems
\textbf{Outside resources:} \smallskip \\ %%% List all resources used OTHER than the textbook, lecture notes, quizzes, worksheets, and previous homework assignments.
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 3 --- Due February 15, 2017 (Wed)
}
\end{center}


\noindent Turn in the following problems from Hammack's book: \\
Chapter 8, problems 2, 8, 12, 14, 18, 20, 22, 24, 28.\\
Chapter 9, problems 6, 8, 14, 16, 26, 34.\\

\noindent{\bf Additional exercises} (to be turned in)

\begin{enumerate}
\item Let $A$ and $B$ be sets.  Prove that $A \cup B = (A - B) \cup (B - A) \cup (A \cap B)$.
\item Prove or disprove each of the following.
\begin{enumerate}
\item For any sets $A$, $B$, and $C$, if $A \cap B = A \cap C$, then $B = C$.
\item For any sets $A$, $B$, and $C$, if $A \cup B = A \cup C$, then $B = C$.
\item For any sets $A$, $B$, and $C$, if $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then $B = C$.
\end{enumerate}
\item Let $A$ and $B$ be sets.  Disprove each of the following.
\begin{enumerate}
\item  \( {\mathcal P}(A\setminus B)\subseteq {\mathcal P}(A) \setminus {\mathcal P}(B)\).
\item  \( {\mathcal P}(A) \setminus {\mathcal P}(B)\subseteq {\mathcal P}(A\setminus B) \).
\item \( {\mathcal P}(A\cup B)\subseteq {\mathcal P}(A) \cup {\mathcal P}(B) \).
\end{enumerate}
\item Let $A$, $B$, and $C$ be sets.  Prove that $(A \cap B) \cup C = A \cap (B \cup C)$ if and only if $C \subseteq A$.
\item Give a precise meaning to the statement ``There is no Universe'' and prove it from the axioms of set theory.
\item Let $A$ be a set.  Determine whether each of the following is or is not a set, according to the axioms of set theory (from Halmos' book or from in-class worksheets).  Justify your answers.
\begin{enumerate}
\item $\{ \{\varnothing, X\} : X \in A \}$
\item $\{ X : X \text{ is a non-empty set }\}$
\item $\{\cP(X) : X \subset A\}$
\end{enumerate}
\end{enumerate}
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