18: Equivalence relations
- Page ID
- 83493
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- 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?