1. Equivalence Relations
Section 7.5
CSE 2813 Discrete Structures

2. Equivalence Relations
A relation on set A is called an equivalence relation if it is:
reflexive
symmetric
transitive
CSE 2813 Discrete Structures

3. CSE 2813 Discrete Structures
Example
Let R be a relation on set A. Is R an equivalence relation?
A = {1,2,3,4,5}
R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(3,1)}
Let R be a relation on set of integers and m is a positive integer > 1. Is R an equivalence relation?
R = {(a,b) | a  b (mod m)}
CSE 2813 Discrete Structures

4. CSE 2813 Discrete Structures
Equivalence Class
Let R be a equivalence relation on set A.
The set of all elements that are related to an element a of A is called the equivalence class of a.
The equivalence class of a w.r.t. R is:
[a]R = {s | (s,a)  R}
When only one relation is under consideration, we will just write [a].
CSE 2813 Discrete Structures

5. CSE 2813 Discrete Structures
Equivalence Example
Consider the equivalence relation R on set A. What are the equivalence classes?
A = {1,2,3,4,5}
R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(3,1)}
CSE 2813 Discrete Structures

6. CSE 2813 Discrete Structures
Partitions
A partition of a set A divides A into non-overlapping subsets
Set A A1 A6 A5 A4 A3 A2
S = {a, b, c, d, e, f }
S1 = {a, d, e}
S2 = {b}
S3 = {c, f }
P = {S1, S2, S3}
P is a partition of set S
CSE 2813 Discrete Structures

7. Partitions and Equivalence Relations
If R is an equivalence relation on set S
then the equivalence classes of R form a partition of S
Conversely, if {Ai | i  I } is a partition of set S,
then there is an equivalence relation R that has the sets Ai (iI) as its equivalence classes
CSE 2813 Discrete Structures

8. CSE 2813 Discrete Structures
Exercises
1, 2, 7, 15, 17, 18, 19, 20, 21, 23, 29
CSE 2813 Discrete Structures