Definition:Equivalence Class

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Definition

Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $x \in S$.


Then the equivalence class of $x$ under $\mathcal R$ is the set:

$\eqclass x {\mathcal R} = \set {y \in S: \tuple {x, y} \in \mathcal R}$


If $\mathcal R$ is an equivalence on $S$, then each $t \in S$ that satisfies $\tuple {x, t} \in \mathcal R$ (or $\tuple {t, x} \in \mathcal R$) is called a $\mathcal R$-relative of $x$.


That is, the equivalence class of $x$ under $\mathcal R$ is the set of all $\mathcal R$-relatives of $x$.


Representative of Equivalence Class

Let $\eqclass x {\mathcal R}$ be the equivalence class of $x$ under $\mathcal R$.

Let $y \in \eqclass x {\mathcal R}$.

Then $y$ is a representative of $\eqclass x {\mathcal R}$.


Notation

The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif $\left[\!\left[{x}\right]\!\right]_\mathcal R$.


Other variants:

  • 1965: Seth Warner: Modern Algebra uses $\bigsqcup_{\mathcal R} \mkern {-28 mu} {\raise 1pt x} \ \ $ for $\eqclass x {\mathcal R}$, which is even more challenging to render in our installed version of $\LaTeX$ than $\eqclass x {\mathcal R}$ itself.


Also known as

The equivalence class of $x$ under $\mathcal R$ can be stated more tersely as the $\mathcal R$-equivalence class of $x$, or just the $\mathcal R$-class of $x$.


The term equivalence set can also occasionally be found for equivalence class.

Some sources, for example 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.), use the term equivalence block.


Examples

Same Age Relation

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { the age of $x$ and $y$ on their last birthdays was the same}$


Then the equivalence class of $x \in P$ is:

$\eqclass x \sim = \set {\text {All people the same age as $x$ on their last birthday} }$


Also see

$y \in \eqclass x {\mathcal R} \iff \paren {x, y} \in \mathcal R$


Technical Note

The $\LaTeX$ code for \(\eqclass {x} {\mathcal R}\) is \eqclass {x} {\mathcal R} .

This is a custom construct which has been set up specifically for the convenience of the users of $\mathsf{Pr} \infty \mathsf{fWiki}$.

Note that there are two arguments to this operator: the part between the brackets, and the subscript.

If either part is a single symbol, then the braces can be omitted, for example:

\eqclass x {\mathcal R}


Sources