diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index 3b39031..5ea992c 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -2917,17 +2917,30 @@ Discuss the result of replacing the union operation by the intersection \section{Exercises 7}% \label{sec:exercises-7} -\subsection{\unverified{Exercise 7.10}}% +\subsection{\partial{Exercise 7.10}}% \label{sub:exercise-7.10} Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive integer $m$ less than $4$. -\begin{proof} +\begin{answer} - TODO + Let $\left< x_1, x_2, x_3, x_4 \right>$ denote an arbitrary $4$-tuple. + Then + \begin{align} + \left< x_1, x_2, x_3, x_4 \right> + & = \left< \left< x_1, x_2, x_3 \right>, x_4 \right> + & \label{sub:exercise-7.10-eq1} \\ + & = \left< \left< \left< x_1, x_2 \right>, x_3 \right>, x_4 \right> + & \label{sub:exercise-7.10-eq2} + \end{align} + Here \eqref{sub:exercise-7.10-eq1} is an equivalent ordered $2$-tuple and + \eqref{sub:exercise-7.10-eq2} is an equivalent ordered $3$-tuple. + Furthermore, $\left< x_1, x_2, x_3, x_4 \right> = + \left< \left< x_1, x_2, x_3, x_4 \right> \right>$, showing it can be + represented as an ordered $1$-tuple as well. -\end{proof} +\end{answer} \section{Functions}% \label{sec:functions}