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EQUIVALENCE RELATIONS
A partition P of a set S is a set of nonempty subsets of S which are pairwise
disjoint and whose union is all of S. An equivalence relation on a set S is a
reflexive, symmetric, transitive relation on S. Both of these structures is motivated
by the idea of grouping objects that are alike in some respect. Partition and
equivalence relation provide exactly the same class of structures. Each definition
is a different way of presenting the same type of structure. This chapter spells this
out in detail.
Contents
Partitions
Definitions
Simple examples
Counting Partitions
Notation
Equivalence relations
Equivalence relations and partitions
An equivalence relation determines a partition
A partition determines an equivalence relation
The fundamental theorem on equivalence relations
Consciousness-raising examples
Answers to exercises
Abstraction of similarity
This section concerns the idea of objects being alike in some respect.
 With poker chips, some are red, some are white and some are blue. You can group
them into three subsets – the red ones, the white ones, and the blue ones. You can say
two red chips are alike in having the same color, and a red and a blue chip are not alike
because they have different colors.
 With natural numbers, some are divisible by 2 and some are not. You can group
them into even ones and odd ones. We can say 4 and 10 are alike in that they are both
even, and 5 and 11 are alike in that they are both odd. However, 4 and 5 are not alike in
that respect.
In both these examples, we can abstract the idea of being alike in some respect by
grouping the like objects together (clumping) or by making a statement that two given
objects are alike or different. (See two.) These two ideas can be abstracted into the
idea of partition and of equivalence relation.
Partitions
2
Definitions
Definition: Partition
If S is a set, a family P of subsets of S (called the blocks or equivalence classes of
P ) is a partition of S if it satisfies the following axioms:
PAR.1 Every block of P is nonempty.
PAR.2 Every element of S is an element of exactly one block of P .
I recommend spending a
little time seeing that the
alternative definition is
equivalent to the first
definition.
Alternative Definition: Partition
If S is a set, a family P of subsets of S is a partition of S if
APAR.1 Every block of P is nonempty.
APAR.2 S is the union of all the blocks in P .
APAR.3 If B and C are blocks in P and B ¹ C , then B Ç C = Æ.
Remarks
 P is the Greek letter capital pi.
 The word “partition” has several other meanings in math.
 In everyday English, the word “partition” can refer to the division of something into
pieces, but it can also refer to a wall that divides a room in two. Don’t let this mess with
your mind: the mathematical idea has nothing to do with walls.
 A partition is a set of blocks. Don’t refer to one block as a partition.
 APAR.3 means by definition that P is pairwise disjoint.
Simple examples
Here are four partitions of the set {1, 2, 3, 4, 5}:
 P 1=



, 3,4}{
, 5}}
{{1,2}{
P 2 = {{1,2,5}{
, 3,4}}
P 3 = {{1,2,3,4,5}}
P 4 = {{1}{
, 2}{
, 3}{
, 4}{
, 5}}
Remarks
Think about the remarks below until you see why they are correct:

, 3,4}}and {{4,3}, {2,1,5}} are the same partition (see here).
{{1,2,5}{
, 3,4}} and {{1,2,4}{
, 5,3}} are different partitions, because they
 {{1,2,5}{
have different blocks.
 In P 2 , as in any partition with two blocks, each block is the complement of the
other one.
 P 3 is the only partition of {1, 2, 3, 4, 5} with exactly one block. Every nonempty
set has a partition consisting of the whole set as the only block.
 P 4 is the only partition with five blocks. Every nonempty set has a partition
whose blocks are all the singleton subsets of the set.
 The empty set has just one partition, whose set of blocks is empty.
Non-examples
3

, 3}}
{{1,2,5}{
is not a partition of the set {1, 2, 3, 4, 5} because the element 4
is not in any block. It is, however, a partition of the set {1, 2, 3, 5}.

{{1,2,4,5},{3,4}} is not a partition (of any set) because 4 is in two different
blocks, violating PAR.1. It violates APAR.3, too, because
{1,2,4,5}¹ {3,4} but
{1,2,4,5}Ç {3,4}¹ Æ.
, 1,2}Ç {3}{
, 3,4}}is not a partition because one of its blocks is emp {{1,2,5}{
ty.
Representing partitions
Small finite partitions can be represented by diagrams like the following, which represents P 1 :
1
2
5
4
3
In this diagram, the circles are merely for grouping. The fact that they are circles has
no significance. All that matters is which elements are in which circles.
Counting Partitions
The functions that count partitions are the Bell numbers and the Stirling numbers of
the second kind.
 The set {1, 2, 3, 4, 5} has 52 different partitions altogether. The Bell number B5 =
52.
 There are 25 different partitions of the set {1, 2, 3, 4, 5} that have three blocks each
( P 1 is one of them). The Stirling number S(5,3) = 25.
Notation
Blocks
x Î S , then there is a unique block of P that contains x as an element. This block is denoted by [ x ]P . For example, [2]P = {1,2,5} ,
2
If P is a partition of a set S and
[3]P 4 = {3} and [3]P 3 = {1,2,3,4,5} .
Sequence notation
It may happen that an author refers to a partition as if it were a sequence of sets.
For example, they might say P 1 is the partition (A1, A2 , A3 ) , where A1 = {1,2} ,
A2 = {3,4} and A3 = {5} . Be warned: the order the blocks are listed in doesn’t
matter, even though in sequence notation it usually does matter.
Exercise PAR
4
Provide an example with the given property of a partition P of the set R of real numbers. Answers.
a) P has at least one block with exactly three elements.
b) {1,2} and {3} are blocks of P .
c) P has at least one finite block and at least one infinite block.
d) P has an infinite number of finite blocks.
e) P has an infinite number of infinite blocks.
Exercise IIZ
Find a partition of Z that has an infinite number of infinite blocks. Answer.
Equivalence relations
The concept of equivalence relation is an abstraction of the idea of two math objects
being like each other in some respect.
 If an object a is like an object b in some specified way, then b is like a in that respect.
 a is like itself in every respect!
So if you want to give an abstract definition of a type of relation intended to capture
the idea of being alike in some respect, two of the properties you could require are reflexivity and symmetry. The nearness relations are reflexive and symmetric. Relations with
those two properties are studied in applied math NEED REFERENCE but here we are
going to require the additional property of transitivity, which roughly speaking forces the
objects to fall into discrete types, making a partition of the set of objects being studied.
Definition: Equivalence relation
An equivalence relation on a set S is a reflexive, symmetric, transitive relation on S.
Note This definition is an abstraction of the idea of “being like each other”. You
don’t have to have some property or mode of similarity in mind to define an equivalence relation.
Examples
See also these examples.
 The relation “equals” on any set is an equivalence relation. On the set {1, 2, 3},
“equals” is the relation {(1,1),(2,2),(3,3)}.
2
2
 The relation E on the real numbers defined by x E y if and only if x = y is an
equivalence relation. In this case, x and y are like each other in this respect: They have
the same square.
 Let A = {1, 2, 3, 4, 5} and let
a = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,2),(2,1),(1,5),(5,1),(2,5),(5,2),(3,4),(4,3)}.
Then a is an equivalence relation on A. You can define this relation more easily by saying:
“ x a y if and only if x and y are in the same block of P 2 (above).” This is the key to the
relationship between equivalence relations and partitions.
 For any integer k, congruence (mod k) is an equivalence relation, discussed here.
Non-examples
5
 The relation “ £ ” on the set of real numbers is not an equivalence relation. It is reflexive and transitive but not symmetric: For example,
3 £ 5 is true but 5 £ 3 is false.
This relation is an example of an ordering.
 The relation E defined on the real numbers by x E y if and only if x = 2 and
y = 2 is not an equivalence relation. It is symmetric and transitive but not reflexive. It is
a partial equivalence relation.
 The relation D defined on a set of sets by x D y if and only if x and y are disjoint is
not an equivalence relation. It is reflexive and symmetric but not transitive.
Equivalence relations and partitions
This is the fundamental fact connecting equivalence relations and partitions:
Every equivalence relation on a set S determines a specific partition of S and
every partition of S determines a specific equivalence relation on S. These operations are inverse to each other.
This statement is not a theorem because I haven’t told you how each equivalence relation induces a partition and vice versa. The proper statement is the Fundamental Theorem on Equivalence Relations below, but to state it requires several definitions. A more
detailed explanation with proofs is given here.
An equivalence relation determines a partition
If an equivalence relation E is given on a set S, the elements of S can be collected together into subsets, with two elements in the same subset if and only if they are related by
E. This collection of subsets of S is a set denoted S/E, the quotient set of S by E. Here is
the formal definition of S/E:
Definition: quotient set of an equivalence relation
x Î S , the equivalence class
of x mod E, denoted by [x ] , is the subset {y Î S | y E x}of S. The quotient set of E
E
Let E be an equivalence relation on a set S. For each
is the set of equivalence classes of x mod E for all
x Î S.
Warning
I have now defined two notations that look like
a partition,
relation,
[ x ]s for some symbol s. If s denotes
[ x ]s is the block of the partition that contains x. If s denotes an equivalence
[ x ]s denotes the set {y Î S | y s x}. Thus the notation “ [ x ]s ” is overloaded.
However, the fundamental theorem below says don’t worry, be happy about this.
Example
Let A = {1, 2, 3, 4, 5} and let
a = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,2),(2,1),(1,5),(5,1),(2,5),(5,2),(3,4),(4,3)}
(We looked at this example before.) Notice that
[2]a = {1,2,5}, because by defini-
tion
[2]a = {y Î A | y a 2}
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and you can see that the only three pairs in a whose second coordinate is 2 are (1,2),
(2,2) and (5,2). Reasoning in the same way, we have
[1]a = [2]a = [5]a = {1, 2,5}and [3]a = [4]a = {3,4}
so by definition the quotient set of a is the partition
In other words,
, 3,4}}, which I called P 2
{{1,2,5}{
above.
A/a = P2.
The fact that the quotient set is a partition is always true:
Theorem
If S is a set and E is an equivalence relation on S, then the quotient set S/E is a partition of S.
Proof
 Proof of Par.1: Any block of S/E is of the form
[ x ] E for some x Î S . By definition,
[ x]E = { y Î S | y E x } . Since E is reflexive, x E x, so [ x ] E is nonempty because it contains x.
 Proof of Par.2: We know that x is in at least one set because x Î [ x ]E by reflexivity. Suppose
x Î [ y ]E and x Î [z]E . We must show that [ y ]E = [z]E as sets. That means by definition of set
equality we must show that for any element w, w Î [ y ]E if and only if w Î [z]E . Suppose (modus ponens) that w Î [ x ]E . Then by comprehension, w E y. Then x Î [ y ]E , so x E y. By symmetry, y E x.
So by transitivity, w E x. Since x Î [z]E , we know x E z. So by transitivity, w E z, which means
w Î [z]E . That is half the proof. You do the other half.
How to think about S/E
If E is an equivalence relation on S, you can think of the quotient set S/E as obtained by merging
equivalent elements of S. In the example above, 3 and 4 are equivalent so they are merged into the
equivalence class {3, 4}, which is both
[3]a and [4]a .
One often says that one identifies equivalent elements. Here, “identify” means “make identical” rather than “discover the identity of”. Mathematicians may also say we glue equivalent elements together.
Thus we form the Möbius strip by gluing the left and right edge of the unit square together in a certain
way.
Exercises
BTWO Let S = {1, 2, 3, 4}. Find two different equivalence relations E and E' with the property that
the subset {1, 2} is a block of both S/E and S/E'. Answer.
INTB Give an example of an equivalence relation E on the set R of real numbers with the property
that the interval [0, 1] is one of the equivalence classes of E. Answer
A partition determines an equivalence relation
EP by the definition:
x EP y if and only if [ x ]P = [y ]P
Given a partition P , you get an equivalence relation
x EP y if and only if x and y are in the same block of P . The relation EP is
well-defined because each element is in exactly one block of P by PAR.2.
In other words,
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Theorem
For a partition P of S, the relation
EP is an equivalence relation on S.
Proof
All three steps in this proof depend on the fact that each element of S is in exactly one block
(Par.2).
 Since x is in the same block as itself, x EP x , so EP is reflexive.
 If
x EP y , then x and y are in the same one and only block, which is the same thing as saying that
y and x are in the same block, so y EP x . So EP is symmetric.
 If x EP y and y EP z , then x is in the same block as y and z is in the same block as y. By Par.2,
x and z must be in the same block, so x EP z Hence EP is transitive.
The fundamental theorem on equivalence relations
One way to state the fundamental theorem
Above we showed how to take an equivalence relation E and construct a partition S/E. Then we
showed how to take a partition P of a set S and construct an equivalence relation EP .
The fundamental theorem on equivalence relations says that if you perform one construction and
then the other you get back what you started with. In other words:
 If you have an equivalence relation E, construct the quotient set S/E, which is a partition, and
then construct the equivalence relation E / SE corresponding to that partition, you get the equivalence
relation E you started with.
 If you have a partition P of S, construct the corresponding equivalence relation
construct the quotient set
EP and then
S / EP of EP , you get the partition P back again.
In short:
ES / E = E
and
S / EP = P
Note This pair of equations condenses the information down to two cryptic statements. It is worthwhile spending a few minutes trying to spell out to yourself in detail just what they say.
Lots of math in research papers is excruciatingly succinct.
You need to practice unwinding what they say.
The proof of the fundamental theorem involves the same sort of arguments given above.
Another way to state the fundamental theorem
Let S be a set, let p (S ) be the set of all partitions of S, and let E(S) be the set of all equivalence relations on S. The fundamental theorem says that the two constructions given above produce a function
R : p (S) ® E (S) and a function Q : E (S ) ® p (S) that are inverses to each other. This means they
are bijections. In fact, E and p are functors that are naturally isomorphic by these bijections.
Example
If S = {1, 2}, then
8
 p (S ) is the set {{{1}, {2}}, {{1, 2}}}. It has two elements, the partition {{1}, {2}} and the partition {{1,
2}}.
 E(S) is the set {{1, 1), (2, 2)}, {(1, 1), (2, 2), (1, 2), (2, 1)}}. It contains the two equivalence relations
{1, 1), (2, 2)} and {(1, 1), (2, 2), (1, 2), (2, 1)}.
 R[{{1}, {2}}] = {(1, 1), (2, 2)} and R[{{1, 2}}] = {(1, 1), (2, 2), (1, 2), (2, 1)}.
 Q[ {(1, 1), (2, 2)} ] = {{1}, {2}} and Q[ {(1, 1), (2, 2), (1, 2), (2, 1)}] = {{1, 2}}.
According to the fundamental theorem,
An equivalence relation is the same thing as a partition
Of course, they are described differently, but once you have an equivalence relation you have a partition and vice versa. All you know about the partition is determined by the data of the equivalence
relation and all you know about the equivalence relation is determined by the data of the partition.
The relationship between equivalence relations and partitions is a basic and important example of
the fact that two different looking kinds of structures can actually be two different descriptions of
the same kind of structure.
Consciousness-raising examples
Partition of Z by remainders To Do
For now, read about this in Wikipedia here.
The quotient of the squaring function
Let P be the partition of R whose blocks are all subsets of the form
{r ,- r }for all r Î R . P
is
2
the quotient set of the squaring function F : R ® R defined by F ( x ) = x . Observe that:
 P has an infinite number of blocks.

{2,- 2}, {p,- p }and {0}are blocks of P .
 Every block of P except {0}has exactly two elements.
A foliation of the real plane
For any given real number c, let
Fc (x) = x2 + c . This defines an infinite family of functions
Fc : R ® R . For a given c, the graph of Fc is the subset (r , r 2 + c ) | r Î R , which I will denote by
{
}
Bc , of the real plane R ´ R . You can check that these subsets Bc are a partition of R ´ R . The picture below shows some of the blocks.
 These blocks are dense in R ´ R , meaning that between any two of them there is an infinite number of others. There is no block next to a given block. They are dense in R ´ R just like the points on
the real line are dense.
 This is a baby example of a foliation (Wikipedia, MathWorld).
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4
3
2
1
-2
-1
1
2
-1
-2
Answers to exercises
Answers to Exercise PAR. Each problem has many possible answers. I give only one for each.
a) Let
B1 = {1,3,5} and B2 = the set of all real numbers except 1, 3 and 5.
b) Let
B1 = {1,2}. B2 = {3} and B3 = the set of all real numbers except 1, 2 and 3.
c) The answers to a) and to b) also work for this one.
d) For each positive integer n, let
Bn = {n,- n}, and let B1 = the set of all real numbers that are
not integers.
e) For each integer n, let
Bn = [n, n + 1) = {x Î R||n £ x < n + 1|}. Can you think of a partition
of the integers Z that has an infinite number of infinite blocks?
Answer to Exercise IIZ
For each prime p, let
set P =
ö
æ
÷
Bp = { pn | n Î N}}
} . Let G = Z - ççç U Bp ÷
÷. (See set subtraction). Then
çèp prime ÷
÷
ø
{Bp | p prime}UG .
Then P is a partition of Z with an infinite number of infinite blocks.
Note: I have deliberately used succinct notation that you may have to unwind extensively before you understand it. (Pretend you are a kitten attacking a ball of yarn.)
10
Answer to Exercise BTWO
There are exactly two such equivalence relations, namely {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)} and
{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,4), (4,3)} . Note that {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (1,3),
(3,1), (2,3), (3,2)} is not an answer. Its partition is {{1, 2, 3}, {4}}, and althouogh 1 and 2 are indeed in the
same block, {1, 2} is not a block.
Answer to Exercise INTB.
The easiest way to answer questions like this is to put everything outside of [0, 1] into singleton
blocks, so the answer would be
xEy if and only if (0 £ x £ 1 and 0 £ y £ 1) or x = y
There is an infinite number of other answers.