Definition 5.36 (Equivalence Classes, Representitives)

Let E be an equivalence relation over a set V.
For every vV we let$$
\color{red} [v]_{E} \coloneqq \set{v' \in V\ |\ (v, v') \in E}

denote the $\color{red}\text{equivalence class of}$ $v$ with respect to $E$ $\color{lightgreen}\text{(i.e. the equivalence class }[V]_{E}\text{ consists of all elements of }V$ $\color{lightgreen}\text{that are }\textquoteleft\text{equivalent}\textquoteright\text{ to }v\text{ according to }E\text{).}$ A set $W \subseteq V$ is an $\color{red}\text{equivalence class}$ (of $E$), if there exists an element $v \in V$ with $W = [v]_{E}$ The element $v$ is then called a $\color{red}\text{representative}$ of its equivalence class $W$ { #def} >[!example] >If $V$ is the set of all people and $E$ is the "having the same age" relation, then an equivalence class $W$ would be the group of all people who are the same age as $v$. This $W$ would be an equivalence class because it contains all individuals who share the same age as person $v$ ### Orders <div class="transclusion internal-embed is-loaded"><a class="markdown-embed-link" href="/leeds-university/computer-science/compulsory-modules/fundamental-math-concepts/5-set-theory/definitions/definition-5-37/" aria-label="Open link"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="svg-icon lucide-link"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a><div class="markdown-embed"> Let $E$ be a binary relation over a set $V$. 1. $E$ is a $\color{red}\text{preorder}$, if $E$ is reflexive ***and*** transitive 2. $E$ is a $\color{red}\text{partial order}$, if $E$ is reflexive, transitive ***and*** antisymmetric 3. $E$ is a $\color{red}\text{linear order}$ or total $\color{red}\text{order}$, if $E$ is reflexive, transitive, antisymmetric ***and*** total </div></div>