7.2 Equivalence Relations

Objectives:

• Construct a type of relation that partitions elements of a set into blocks

• Utilize these categories to renew our understanding of modular arithmetic

7.2.1 A Stronger Relation

Example 7.2.1.

Consider the relation on $\mathbb{Z}$ given by $a\sim b$ if $5$ divides $b-a$. Let $[a]:=\{x\in \mathbb{Z}:a\sim x\}$. Describe all unique sets $[a]$ (like $[0]$, $[1]$,…). Draw a diagram showing all of these sets. Do these sets have any intersections, or are they all separate? Are there any integers that aren’t in a set $[a]$ or are in a set by themselves?

Blank space for you to write your work.

In the next few sections we describe another special type of relation where the following diagrams are possible. Not all relations have this well-partitioned picture; they have to satisfy the conditions of an equivalence relation.

Definition 7.2.2.

An equivalence relation on $X$ is a relation (that is, a subset $S$ of $X\times X$, where $(x,y)\in S\iff x\sim y$) with the following properties:

1. (Reflexivity) $x\sim x$ for each $x\in X$.

2. (Symmetry) Whenever $x\sim y$ for $x,y\in X$, then $y\sim x$.

3. (Transitivity) Whenever $x\sim y$ and $y\sim z$ for $x,y,z\in X$, then $x\sim z$.

Example 7.2.3.

Let ${\mathbb{Z}}^{\ast }:=\mathbb{Z}\setminus \{0\}$. Define the relation $\sim$ on $\mathbb{Z}\times {\mathbb{Z}}^{\ast }$ by, for all $a,c\in \mathbb{Z}$ and all $b,d\in {\mathbb{Z}}^{\ast }$,

$$(a,b)\sim (c,d)\text{iff}ad=bc.$$

Show that $\sim$ is an equivalence relation.

Blank space for you to write your work.

Given our understanding that relations of this form are similar to how we say two fractions represent the same quantity, this helps us understand why these are called equivalence relations: any two related elements can be considered equivalent.

Example 7.2.4.

Consider the relation on $ℝ$. Show that is not an equivalence relation. Then explain why any two related elements cannot be considered equivalent.

Blank space for you to write your work.

7.2.2 Equivalence Classes

We now formalize the notion of what it means for two elements in an equivalence relation to be truly “equivalent”.

Definition 7.2.5.

Let $\sim$ be an equivalence relation on a nonempty set $X$, and let $a\in X$. The equivalence class of $a$ is the set

$$[a]:=\{x\in X\mid x\sim a\}.$$

The set of all equivalence class of $\sim$ is denoted by

$$X/\sim :=\{[a]\mid a\in X\}.$$

For example, in Example 7.1.3, $[(1,2)]=\{\mathrm{\dots },(-3,-6),(-2,-4),(-1,-2),(1,2),(2,4),(3,6)\mathrm{\dots }\}$. This equivalence class is typically referred to as $\frac{1}{2}$ - even without the brackets, it is known to be equivalent to $\frac{2}{4}$, $\frac{-4}{-8}$, and all other elements in its equivalence class.

Theorem 7.2.6.

Let $\sim$ be an equivalence relation on a set $A$ and let $a,b\in A$. Then $[a]=[b]$ iff $a\sim b$.

Blank space for you to write your work.

The following theorem states that the equivalence classes of $A$ are distinct from one another (part (b)) and that every element of $A$ is in its own equivalence class (part (a)).

Theorem 7.2.7.

Suppose $\sim$ is an equivalence relation on a set $A$. Then

1. ${\bigcup }_{a\in A}[a]=A$.

Blank space for you to write your work.

2. for all $a,b\in A$, either $[a]=[b]$ or $[a]\cap [b]=\mathrm{\varnothing }$.

   Blank space for you to write your work.