Properties:: [[two equivalence classes are either disjoint or equal]], [[partitions are always determined uniquely by equivalence relations]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[equivalence class]]) > Let $\sim$ be an [[equivalence relation]] on a set $A$. We define the **equivalence class determined (or 'represented') by** $x \in A$ as $[x]:=\{ y \in A : y \sim x \}.$ Note that via reflexivity of $\sim$, $x \sim x$. \ The [[quotient set|quotient]] of $A$ by $\sim$, denoted by $A / {\sim}$, is the set $\{ [x]: x \in A \}$. > Note that any pair of equivalence classes are either disjoint or equal. Indeed, if $[x] \cap [y] \neq \emptyset$, then there exists $z$ such that $x \sim z \sim y$; hence $x \sim y$. > [!basicexample] > - Define two points in the plane to be equivalent if they lie at the same ([[euclidian metric|Euclidean]]) distance from the origin. Then the [[equivalence class]]es are concentric circles centered around the origin as well as the set consisting of the origin alone. > - Let $\mathscr{L}$ be the collection of all [[line]]s in the parallel to the line $y=-x$. Then $\mathscr{L}$ is a [[partition]] of the plane, since each point lines on exactly on line. Each line is an [[equivalence class]] of the [[equivalence relation]] defining two points to be equivalent if they lie on the same line parallel to $y=-x$ (see [[partitions are always determined uniquely by equivalence relations]].) Compare to the [[quotient operator]] on a [[vector space]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```