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Enderton. Exercise set 7.

finite-set-exercises
Joshua Potter 2023-06-23 06:48:29 -06:00
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@ -2917,17 +2917,30 @@ Discuss the result of replacing the union operation by the intersection
\section{Exercises 7}% \section{Exercises 7}%
\label{sec:exercises-7} \label{sec:exercises-7}
\subsection{\unverified{Exercise 7.10}}% \subsection{\partial{Exercise 7.10}}%
\label{sub:exercise-7.10} \label{sub:exercise-7.10}
Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
integer $m$ less than $4$. 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}% \section{Functions}%
\label{sec:functions} \label{sec:functions}