Equivalence relation: Difference between revisions

Revision as of 13:34, 25 March 2026

An equivalence relation is a binary relation on a set that groups elements into categories^[1] in which all members are considered "equivalent" under some criterion.

Definition

A relation $\sim$ on set $X$ is a equivalence relation if it satisfies the following properties:

Reflexivity: $\forall x \in X$ , $x \sim x$ .
Symmetry: $\forall x, y \in X$ such that $x \sim y$ , $y \sim x$ .
Transitivity: $\forall x, y, z$ , if $x \sim y$ and $y \sim z$ , then $x \sim z$ .

When $x \sim y$ , " $x$ is said to be equivalent to $y$ " under the relation $\sim$ .

Equivalence classes

Given $x \in X$ , the equivalence class of $x$ , denoted $[x] = {y \in X ∣ y \sim x}$ is the set of elements that are equivalent to $x$ .

Notes

↑ Not to be confused with category in category theory.

[1] Not to be confused with category in category theory.

[1]

@@ Line 8: / Line 8: @@
 * '''Transitivity''': <math>\forall x,y,z</math>, if <math>x\sim y</math> and <math>y\sim z</math>, then <math>x\sim z</math>.
-When <math>x\sim y</math>, then we say that "<math>x</math> is equivalent to <math>y</math>" under the relation <math>\sim</math>.
+When <math>x\sim y</math>, "<math>x</math> is said to be equivalent to <math>y</math>" under the relation <math>\sim</math>.
+== Equivalence classes ==
+Given <math>x \in X</math>, the '''equivalence class''' of <math>x</math>, denoted <math display="block">[x]= \{y \in X \mid y \sim x\}</math> is the set of elements that are equivalent to <math>x</math>.
 == Notes ==
 <references group="Note" />