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\begin{document}

\title[Math 250 Handout 6 - Sets]{Math 250 Handout 6 - Sets}\maketitle

\begin{enumerate}
\item Recall that $d\mathbf{Z} = \{n : n \in \mathbf{Z} \text{ s.t. } d \mid n
 \}$.
 \begin{enumerate}
 \item $25\mathbf{Z} \subset 5 \mathbf{Z}$;
 \item $5\mathbf{Z} \subset 25\mathbf{Z}$;
 \item $24\mathbf{Z} \subset 4 \mathbf{Z}$; 
 % \item $n\mathbf{Z} \subset m\mathbf{Z}$ if and only if $m \mid n$.

 \end{enumerate}
 \vspace{10pt}

\item Prove or disprove each of the following:
 \begin{enumerate}
 \item $(-1,1) \subset (-2,2)$. 
 \item $(-1,2) \subset (-2,1)$.
 \end{enumerate}
 \vspace{10pt} 

 
\item Let $A,B,C$ and $D$ be arbitrary sets. Prove or disprove the following.
 \begin{enumerate}
 \item If $A \subset B$, $B \subset C$, and $C \subset D$, then $A \subset D$.
 \item If $A \not \subset B$ and $B \not \subset C$, then $A \not \subset C$.
 \item If $A \subset B$ and $B \not \subset C$, then $A \not \subset C$.
 \end{enumerate}
\vspace{10pt}

\item Prove each of the following:
 \begin{enumerate}
 \item $(-10,5] \cap [0,10] = [0,5]$.
 \item $(-10,5] \cup [0,10] = (-10,10]$.
 \item $(-10,5] - [0,10] = (-10,0)$.
 \end{enumerate}

\vspace{10pt}

\item Prove that $4\mathbf{Z} - 6\mathbf{Z} = 4\mathbf{Z} - 3\mathbf{Z}$.
\vspace{10pt}

\item Let $A,B \subset C$ be sets. Prove each of the following:
 \begin{enumerate}
 \item $A \cap B \subset A$;
 \item $A \cap \emptyset = \emptyset$;
 \item Suppose that $B \subset C$. Prove that $A-C \subset A-B$.
 \item $A \subset B$ if and only if $A \cap B = A$.
 \end{enumerate}

\end{enumerate}

\end{document}

