%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% %2 %%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Prove that if $A \subseteq B$ and $B \subseteq A$, then $A=B$.} \begin{remark*} Combined with Proof \ref{pf:1}, these two statements yield the following theorem. \end{remark*} \begin{thm}[Double containment] \label{th:dc} If $A$ and $B$ are sets, then $A=B$ if and only if $A \subseteq B$ and $B \subseteq A$. \end{thm} The ``if and only if" means that the statement can be read in either direction. Specifically, ``$A=B$ if and only if $A \subseteq B$ and $B \subseteq A$" is equivalent to the following statements: \begin{enumerate}[label={(\alph*)}] \item if $A = B$, then $A \subseteq B$ and $B \subseteq A$, and \item \label{dc:2} if $A\subseteq B$ and $B \subseteq A$, then $A = B$. \end{enumerate} This result (in particular, statement \ref{dc:2}) is one of the most common methods for showing that two sets are equal. Since this result relies on showing that each of the two sets is contained within the other, we call this result the Double Containment (or Double Inclusion) Principle. \section{First draft} Due Friday, February 12 at 6:00 PM. \begin{proof} Write your proof here. \end{proof}