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Version 2 |
| Let $S$ be a set with an equivalence relation $\sim$. An {\em equivalence class} of $S$ under $\sim$ is a subset $T\subset S$ such that |
Let $S$ be a set with an equivalence relation $\sim$. An {\em equivalence class} of $S$ under $\sim$ is a subset $T\subset S$ such that |
| \begin{itemize} |
\begin{itemize} |
| \item If $x \in T$ and $y \in S$, then $x \sim y$ if and only if $y \in T$ |
\item If $x \in T$ and $y \in S$, then $x \sim y$ if and only if $y \in T$ |
| \item If $S$ is nonempty, then $T$ is nonempty |
\item If $S$ is nonempty, then $T$ is nonempty |
| \end{itemize} |
\end{itemize} |
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The {\em set of all equivalence classes} of $S$ under $\sim$ is defined to be the set of all subsets of $S$ which are equivalence classes of $S$ under $\sim$. |
| For $x \in S$, the equivalence class containing $x$ is often denoted by $[x]$, so that |
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| $$ |
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| [x] := \{ y \in S \mid x \sim y \}. |
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| $$ |
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| The {\em set of all equivalence classes} of $S$ under $\sim$ is defined to be the set of all subsets of $S$ which are equivalence classes of $S$ under $\sim$, and is denoted by $X/\sim$. The map $x\mapsto [x]$ is sometimes referred to as the \PMlinkescapetext{canonical projection}. |
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| For any equivalence relation $\sim$, the set of all equivalence classes of $S$ under $\sim$ is a partition of $S$, and this correspondence is a bijection between the set of equivalence relations on $S$ and the set of partitions of $S$ (consisting of nonempty sets). |
For any equivalence relation $\sim$, the set of all equivalence classes of $S$ under $\sim$ is a partition of $S$, and this correspondence is a bijection between the set of equivalence relations on $S$ and the set of partitions of $S$ (consisting of nonempty sets). |
