diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index d3965c6..3fc6896 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -31,6 +31,12 @@ For any relation $R$ there is a function $H \subseteq R$ with $\dom{H} = \dom{R}$. +\section{\partial{Axiom of Choice, Second Form}}% +\label{ref:axiom-of-choice-2} + +For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$ + for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$ + \section{\defined{Composition}}% \label{ref:composition} @@ -3929,4 +3935,19 @@ Define \end{proof} +\section{Exercise 9}% +\label{sec:exercise-9} + +\subsection{\unverified{Exercise 9.31}}% +\label{sub:exercise-9.31} + +Show that from the first form of the axiom of choice we can prove the second + form, and conversely. + +\begin{proof} + + TODO + +\end{proof} + \end{document} diff --git a/preamble.tex b/preamble.tex index 27ca766..4a6203e 100644 --- a/preamble.tex +++ b/preamble.tex @@ -6,7 +6,7 @@ \usepackage{environ} \usepackage{fancybox} \usepackage{fontawesome5} -\usepackage{mathrsfs} +\usepackage{mathabx, mathrsfs} \usepackage{soul} \usepackage{stmaryrd} \usepackage[usenames,dvipsnames]{xcolor}