discrete mathematics for engineering . matrix representation by sushil kumar

1. CENTURION UNIVERSITY OF
TECHNOLOGY &MANAGEMENT
PROJECT :- 3
Prepared By:-
Sushil kumar (220101120119)
Guided By :-
Dr. Ashok Mishra
Properties Of A Relation On A Set Using The Matrix
Representation With Examples
Discrete mathematics

2. Representing Relation
Using Matrix
A relation between two non-empty sets can be represented
using adjacency matrix.
 Suppose R is a relation from A = {a1 , a2 , …, am } to B = {b1 ,
b2 , …, bn }.
 The relation R is represented by the matrix MR = [mij],
The matrix representing R has a 1 as its (i,j) entry when ai
is related to bj and a 0 if ai is not related to bj .

3. Reflexive Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Reflexive
If and only if,
( x,x ) ∈ R, “∀ x ∈ A
Reflexive relation is a relation of elements of a set A
such that each element of the set is related to itself.

4. Reflexive Relation in a
adjacency Matrix
Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
mii = 1 ∀ i.
Let ,
Set A= {1,2,3}
Relation “R” is reflexive relation
define on set A
R= {(1,1),(2,2),(3,3)}
Example 1 2 3
1
2
3
1
1
1
0
0 0
0
0 0
MR =

5. Irreflexive Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Irreflexive
If and only if,
( x,x ) ∉ R, “∀ x ∈ A
Reflexive relation is a relation of elements of a set A such
that ∃ no any element of the set is related to itself.

6. Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
Irreflexive Relation in a
adjacency Matrix
mii = 0 ∀ i.
Example
Let ,
Set A= {1,2,3}
Relation “R” is irreflexive relation
define on set A
R= {(1,3),(2,1),(2,3),(3,2)}
1 2 3
1
2
3
0
0
0
0
0
1
1
1
1
MR =

7. Symmetric Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Symmetric iff,
( x,y ) ∈ R ⇒ ( y,x ) ∈ R, “∀ x,y ∈ A
A symmetric relation between two or more elements of a set is such that
if the first element is related to the second element, then the second
element is also related to the first element as defined by the relation.

8. Symmetric Relation in a
adjacency Matrix
Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
mij =1, then mji =0, “∀ ji
mji =0, then mij =1, “∀ ji
Example
Let ,
Set A= {1,2,3}
Relation “R” is Symmetric relation
define on set A
R= {(1,2),(2,1),(3,1),(1,3),(2,3),(3,2),(1,1)}
1 2 3
1
2
3
0
0
1
MR =
1 1
1
1
1 1

9. Assymmetric Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Assymmetric iff,
( x,y ) ∈ R ⇒ ( y,x ) ∉ R, “∀ x,y ∈ A
A symmetric relation between two or more elements of a set is such that
if the first element is related to the second element, then the second
element is not related to the first element as defined by the relation.

10. Assymmetric Relation
in a adjacency Matrix
Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
mji =0 ⇒ m ij = 1 , “∀ ji
Example
Let ,
Set A= {1,2,3}
Relation “R” is Assymmetric
relation define on set A
R= {(1,2),(3,1),(2,3)}
1 2 3
1
2
3
0
0
MR =
1
1
0
1 0
0
0

11. Antisymmetric Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Antisymmetric iff,
( x,y ) ∈ R ∧ ( y,x ) ∈ R ⇒ x=y , “∀ x,y ∈ A
A symmetric relation between two or more elements of a set is such that
if the first element is related to the second element and the second
element also related to the first element, Then first element is equal to
second as defined by the relation.

12. Antisymmetric Relation
in a adjacency Matrix
Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
mij =1 ⇒ mji = 0, “∀ij
except for some [ i=j ]
Example
Let ,
Set A= {1,2,3}
Relation “R” is Assymmetric
relation define on set A
R= (1,2),(3,1),(2,2),(3,3)
1 2 3
1
2
3
MR =
1
1
1
1 0
0
0
0 0

13. Transitive Relation
Let,
A be a non-empty set,
R is a relation define on set A
Relation R is Transitive iff,
( x,y ) ∈ R ∧ ( y,z ) ∈ R ⇒ (x=z) ∈ R , “∀ x,y,z ∈ A
Transitive relations are relations defined on a set such that if the first
element is related to the second element, and the second element is
related to the third element of the set, then the first element must be
related to the third element.

14. Transitive Relation in a
adjacency Matrix
Let R be a relation on a set and
let M be its adjacency matrix.
R is reflexive iff.
mij =1, mjk =1 ⇒ mik = 1, “∀ijk
Example
Let ,
Set A= {1,2,3}
Relation “R” is Symmetric
relation define on set A
R= (1,2),(2,3),(3,1)
1 2 3
1
2
3
MR =
1 1
1
0
0
0
0
0
0