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9
Advanced/Quotient Groups/parts/0 mod.tex
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@ -1,7 +1,7 @@
\section{Modular Arithmetic} \section{Modular Arithmetic}
I'm sure you're all familiar with 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 $n$. \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{} \problem{}
Let $A$ be a set and $\sim$ an equivalence relation. \par 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?} \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 \vfill