diff --git a/Advanced/Quotient Groups/parts/0 mod.tex b/Advanced/Quotient Groups/parts/0 mod.tex index 7885b8a..67ff605 100644 --- a/Advanced/Quotient Groups/parts/0 mod.tex +++ b/Advanced/Quotient Groups/parts/0 mod.tex @@ -1,7 +1,7 @@ \section{Modular Arithmetic} I'm sure you're all familiar with modular arithmetic. -In this section, our goal is to meet \textit{equivalence relations}, +In this section, our goal is to define \textit{equivalence relations}, \textit{equivalence classes}, and use them to formally define arithmetic in mod $n$. @@ -88,14 +88,9 @@ What is the equivalence class of $3$ in $\mathbb{Z}$ under $\equiv_5$? \par \problem{} Let $A$ be a set and $\sim$ an equivalence relation. \par -Show that every element of $A$ is in \textit{exactly one} equivalence class\footnotemark{}\hspace{-1ex}. \par +Show that every element of $A$ is in \textit{exactly one} equivalence class. \par \hint{What properties does an equivalence relation satisfy?} -\footnotetext{ - We could also say \say{$A$ is partitioned by $[A ~/ \sim]$} - or \say{$A$ is the disjoint union of $[A ~/ \sim]$,} \par - where $[A ~/ \sim]$ is the set of equivalence classes of $\sim$. -} \vfill