This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Equivalence relations versus partitions
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of December 6, 2017

2. Definition
An equivalence (relation) on X is any binary relation that is reflexive on X,
symmetric on X, and transitive on X.
Informal Example
1. Let us say that Person1 is name-related to Person2
if they have the same first name. This is an equivalence.
2. Let us say that Person1 is birthday-related to Person2
if they have birthdays on the same day of a year (not necessarily in the same year).
This is an equivalence.
3. Let us say that Person1 is language-related to Person2
if they can speak the same language.
This is not an equivalence because it is not transitive – consider this situation:
Adam: only English, Betty: English and Spanish, Charles: only Spanish.
Exercise. Check if =Z, =Z, <Z, >Z, Z, Z, 1Z2 and | are equivalence relations.

3. Example 1.
1. =X is an equivalence on X.
2. ≡k is an equivalence on Z (for any fixed k).
3. The empty relation ∅ is an equivalence on ∅.
4. 1X2 , the total binary relation on X, is an equivalence on X.
5. If sets X, Y are disjoint then 1X2 ∪ 1Y 2 is an equivalence on X ∪ Y .
6. { 0, 0 ,
1, 1 , 1, 2 , 2, 1 , 2, 2 ,
3, 3 , 3, 4 , 3, 5 , 4, 3 , 4, 4 , 4, 5 , 5, 3 , 5, 4 , 5, 5 }
is an equivalence on {0, 1, 2, 3, 4, 5}.
7. Let X be a set of straight lines on a plane.
The relation || (of being parallel) is an equivalence on X.

4. Fact
1. Let f be a function on a set X.
Consider the following relation over X: xRy iff f(x)=f(y).
This relation is an equivalence on X.
2. Let {Xi : i ∈ I} is a disjoint family of sets.
then i∈I 1Xi
2 is an equivalence on i∈I Xi .
Exercise
1. For each equivalence relation R from Example 1 above specify a function f on a
set a set X such that xRy iff f(x)=f(y).
2. For each equivalence relation R from the Example 1 above specify a disjoint
family of sets {Xi : i ∈ I} such that R = i∈I 1Xi
2 .

5. Definition
Let R be an equivalence on X.
1. Let a ∈ X.
The equivalence class of R determined by a is [a]R = {b ∈ X : aRb} .
2. E is an equivalence class of R if there exists a ∈ X s.t. E = [a]R .
3. Let E be an equivalence class of R.
a is a representative of the equivalence class E of R
if E = [a]R.
4. The quotient (set) of X by R is X/R = {[a]R : a ∈ X}.
Note
1. So, [a]R, or just [a], the equivalence class determined by a,
is the set of all elements which are related to a.
2. So, X/R, the quotient of X by R,
is a family of all equivalence classes of R.

6. Exercise. Investigate equivalence classes of each relation from Example 1 above.
Specify the set X such that the relation is an equivalence on X.
For every element of X specify the equivalence class this element determines.
How many equivalence classes are there?
Are the equivalence classes pairwise disjoint?
Do the equivalence classes form a partition of X?

7. Example
≡3 is an equivalence on Z.
[0] contains integers divisible by 3, i.e. those that have remainder 0 when divided by 3.
[1] contains integers that have remainder 1 when divided by 3.
[2] contains integers that have remainder 2 when divided by 3.
[3] = [0], [4] = [1], [5] = [2], etc.
Z/ ≡3 = {[0], [1], [2]}
Equivelence classes [0], [1], [2] are non-empty, pairwise disjoint and cover Z.
So, Z/ ≡3 is a partition of Z.
In general, every equivalence on X determines a similar way into a partition of X.
Equivalence R on X determines partition PR of X.

8. Example
Let X0 = {n ∈ Z : ∃k n = 3k}.
Let X1 = {n ∈ Z : ∃k n = 3k + 1}.
Let X2 = {n ∈ Z : ∃k n = 3k + 2}.
X0, X1, X2 are non-empty, pairwise disjoint and cover Z.
So, {X0, X1, X2} is a partition of Z.
Define a relation R over Z such that mRn iff m, n belong to the same set Xi.
So, mRn iff m, n have the same remainder when divided by 3.
So, mRn iff m ≡3 n.
So, R is ≡3.
In general, every partition of X determines a similar way an equivalence on X.
Partition P of X determines equivalence RP on X.