Equivalence Relations
MSU CSE 260
Outline
• Introduction
• Equivalence Relations
– Definition, Examples
• Equivalence Classes
– Definition
• Equivalence Classes and Partitions
– Theorems
– Example
Introduction
• Consider the relation R on the set of MSU
students:
a R b  a and b are in the same graduating class.
– R is reflexive, symmetric and transitive.
• Relations which are reflexive, symmetric and
transitive on a set S, are of special interest because
they partition the set S into disjoint subsets, within
each of which, all elements are all related to each
other (or equivalent.)
Equivalence Relations
• Definition. A relation on a set A is called an
equivalence relation if it is reflexive,
symmetric, and transitive.
• Two elements related by an equivalence
relation are called equivalent.
Example
• Consider the Congruence modulo m relation
R = {(a, b)  Z | a  b (mod m)}.
– Reflexive. a  Z a R a since a - a = 0 = 0  m
– Symmetric. a, b  Z If a R b then a - b = km.
So b - a = (-k) m. Therefore, b R a.
– Transitive. a, b, c  Z If a R b  b R c then a - b = km
and b - c = lm. So (a-b)+(b-c) = a-c = (k+l)m. So a R c.
 R is then an equivalence relation.
Equivalence Classes
• Definition. Let R be an equivalence relation
on a set A. The set of all elements related to
an element a of A is called the equivalence
class of a, and is denoted by [a]R.
• [a]R = {xA | (a, x)  R}
• Elements of an equivalence class are called
its representatives.
Example
• What are the equivalence classes of 0, 1, 2, 3…
for congruence modulo 4?
–
–
–
–
[0]4 = {…, -8, -4, 0, 4, 8, …}
[1]4 = {…, -7, -3, 1, 5, 9, …}
[2]4 = {…, -6, -2, 2, 6, 10, …}
[3]4 = {…, -5, -1, 3, 7, 11, …}
The other equivalence classes are identical to one of the above.
[a]m is called the congruence class of a modulo m.
Equivalence Classes & Partitions
• Theorem. Let R be an equivalence relation
on a set S. The following statements are
logically equivalent:
–aRb
– [a] = [b]
– [a][b]  
Equiv. Classes & Partitions - cont
• Definition. A partition of a set S is a collection
{Ai | i  I} of pairwise disjoint nonempty subsets
that have S as their union.
– i,jI Ai  Aj = , and iI Ai = S.
• Theorem. Let R be an equivalence relation on a
set S. Then the equivalence classes of R form a
partition of S.
Conversely, for any partition {Ai | i  I} of S there
is an equivalence relation that has the sets Ai as its
equivalence classes.
Example
• Every integer belongs to exactly one of the four
equivalence classes of congruence modulo 4:
–
–
–
–
[0]4 = {…, -8, -4, 0, 4, 8, …}
[1]4 = {…, -7, -3, 1, 5, 9, …}
[2]4 = {…, -6, -2, 2, 6, 10, …}
[3]4 = {…, -5, -1, 3, 7, 11, …}
• Those equivalence classes form a partition of Z.
– [0]4  [1]4  [2]4  [3]4 = Z
– [0]4, [1]4, [2]4 and [3]4 are pairwise disjoint.