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 $\text{[math]}$ on set $\text{[math]}$ is a equivalence relation if it satisfies the following properties:

Reflexivity: $\text{[math]}$ , $\text{[math]}$ .
Symmetry: $\text{[math]}$ such that $\text{[math]}$ , $\text{[math]}$ .
Transitivity: $\text{[math]}$ , if $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$ .

When $\text{[math]}$ , " $\text{[math]}$ is said to be equivalent to $\text{[math]}$ " under the relation $\text{[math]}$ .

Equivalence classes

Given $\text{[math]}$ , the equivalence class of $\text{[math]}$ , denoted

$\text{[math]}$

is the set of elements that are equivalent to $\text{[math]}$ .

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" />