18: Equivalence relations
- Page ID
- 83493
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- 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?