MAT 2720
Discrete Mathematics
Section 3.3
Relations
http://myhome.spu.edu/lauw
Goals

Relations
• Properties of Relations on X
Recall

A relation from X to Y is a subset
R  X Y



Sometimes, we write
xRy if  x, y   R
Domain of R = all possible value of x
Range of R = all possible value of y
Recall

A relation from X to X is called a
relation on X
R XX
Properties of Relation on X
R is…
Reflexive
Symmetric
Transitive
If…
Diagraph
Example 5(a)
X  {1, 2,3, 4}; R  X  X
R  1,1 , 1,3 ,  2, 2  ,  3, 4 
Reflexive?
Example 5(b)
X  {1, 2,3, 4}; R  X  X
R  1,1 , 1,3 ,  2, 2  ,  3, 4 
Symmetric?
Example 5(c)
X  {1, 2,3, 4}; R  X  X
R  1,1 , 1,3 ,  2, 2  ,  3, 4 
Transitive?
Properties of Relation on X
R is…
Antisymmetric
If…
Diagraph
(Read)
MAT 2720
Discrete Mathematics
Section 3.4
Equivalence Relations
http://myhome.spu.edu/lauw
Goals


Equivalence Relations
• A special relation with nice properties.
• Partition of sets (Clumping Property).
• Applications to counting problems.
CS students should read the
applications in p.166-168
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
Example
( , ) R
( , ) R
( , ) R
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
Reflexive?
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
Symmetric?
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
Transitive?
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
“Clumping” Effective
Definitions and Notations
is an Equivalence Relation if R
is reflexive, symmetric, and transitive.
 R XX
Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
Show that R is an Equivalence Relation
Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
Show that R is an Equivalence Relation
3 divides x  y

Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
Proof: Reflexive
Analysis
Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
Proof: Symmetric
Analysis
Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
Proof: Transitive
Analysis
Definitions and Notations
is an Equivalence Relation if R
is reflexive, symmetric, and transitive.
Equivalence Class of a  X :
 a   x | xRa
  x |  x, a   R
 R XX

Example 1
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
[1]  
[2]  
[3]  
 [1] 
 [2] 
 [3] 
Observations
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
[1]  
[2]  
[3]  
1.
2.
 [1] 
 [2] 
 [3] 
Observations
X  1, 2,3, 4,5, 6, 7,8,9,10 , R  X  X
 x, y   R if 3 divides x  y
[1]  
[2]  
[3]  
1.
2.
 [1] 
 [2] 
 [3] 
1
4
7
10
2
8
5
6
3
9
X
Partition of a Set (1.1)

A partition of a set X is a way to split X
into the union of disjoint subsets.
S   A, B, C, D, E
B
A
E
C
D
X
Partition of a Set (1.1)


A partition of a set X is a way to split X
into the union of disjoint subsets.
For every element in X,
S   A, B, C, D, E
it belongs to one and only one
subset in the partition.
B
A
E
C
D
X
Theorem
Let R  X  X be an equivalence relation.
Then S  [a] | a  X  forms a partition on X .
b
a
a
b 
c
c
X
“Informal” Example
X  The 30 days of April of 2008 ,
R XX
( x, y )  R if x and y are on the
same "position" of the week.
Partition
1  1,8,15,22,29 5  5,12,19,26
 2  2,9,16,23,30 6  6,13,20,27
3  3,10,17,24 7  7,14,21,28
 4  4,11,18,25
Theorem
Let S be a partition on X , define R  X  X by
 x, y   R if x, y belong to the same subset in S ,
then R is an equivalence relation.
S   A, B, C, D, E
B
y
x
A
E
C
D
X
Example 2
X  1, 2,3, 4,5, 6 ,
A  1 , B  2, 4 , C  3,5, 6
1
2
3
S  { A, B, C}
4
5
6
R
(It is easy to check that R is an equivalence relation.)
Summary of the 2 Theorems
Partition
Equivalence Relation
Theorem
Let R be an equivalence relation on a finite set X ,
If each equivalence class has r elements,
then there are
X
r
equivalence class
Theorem
Let R be an equivalence relation on a finite set X ,
If each equivalence class has r elements,
then there are
X
r
equivalence class
X
A1
A2
A3
Ak