18: Equivalence relations Last updated Save as PDF Page ID 83493 Jeremy Sylvestre University of Alberta Augustana 18.1: Motivation There are often situations where we want to group certain elements of a set together as being “the same.” 18.2: Basics and Examples What properties should a relation on a set have to be useful as a notion of “equivalence”? 18.3: Classes, partitions, and quotients As desired (see Section 18.1), an equivalence relation can be used to group equivalent objects together. 18.4: Important examples Equality is the strongest form of equivalence. The “strongest” equivalence relation on a set A is the identity relation, where a≡b if and only if a=b. In this case, each equivalence class is a singleton: [a]={a} for each a∈A. 18.5: Graph for an equivalence relation Given an equivalence relation on a finite set A, what will we observe if we draw the relation's graph? 18.6: Activities 18.7: Exercises