This document defines properties of relations on a set using matrix representation. It provides examples of defining relations between sets using subsets of the Cartesian product. Relations can be represented by matrices where the (i,j)th entry is 1 if the element is in the relation and 0 if not. The properties of reflexive, symmetric, anti-symmetric, and transitive are defined in terms of patterns in the relation matrix. Examples of relation matrices demonstrating each property are given.

1. DISCRETE MATHEMATICS
PROJECT NO. (3)
NAME :- kanta anuraga sahoo
REGD. NO. :- 220301120178
SECTION :- C
BRANCH :- B .Tech ( CSE )

2. PROJECT NAME
DEFINE THE PROPORTIES
OF A RELATION ON A SET
USING THE MATRIX
REPRESENTATION OF THAT
RELATION WITH EXAMPLES

3. RELATION :-
If ‘a’ and ‘b’ are two sets we define a binary
relation from ‘a’ to ‘b’ in any relation i.e.
R⊆(A×B) .
Where ‘A’ is the subset of A×B i.e. A⊆(A×B) .
Ex. :- Let A={1,2}
B={3,4}
A×B = { (1,3),(1,4),(2,3),(2,4) }
R1={ (1,3),(2,3) }
Here , R1⊆ (A×B)
1
2
3
4
A B

4. REPRESENTATION OF RELATIONS BY USING MATRICES :-
We know that if R⊆A×B , then it is a relation .
Any relation ‘R’ can be represented in form of a matrix as “ MR ”, where MR=
[Mij]m×n .
Where , Mij= { 1, if (ai,bj)∈R 0, if (ai,bj)∉R }
Let A={a1,a2 …….. am}
B={b1, b2 …….bn }
A×B ={ (ai,bj) : i=1,2…….m , j=1,2,……n}
 Ex:- Let A={1,2,3} , B={ a,b,c }
define a relation R=[ (1,a),(1,c),(2,b),(2,c),(2,a),(3,a),(3,c) }
i.e. MR=[Mij]3×3=
1 0 0
1 1 1
1 0 1

5. Example :-
2
Let A={ a,b,c,d }
i.e. R={ (a,c),(b,d),(c,a),(a,d),(c,b),(b,b),(d,b),(a,b) }
MR=[Mij}4×1 =
0 1 1 1
0 1 0 1
1 1 0 0
0 1 0 0
It is a relation over a single set , where MR=[Mij]m×m means if a set ‘A’
is there then a relation “R” on set A is the subset of “A×A” .

6. PROPORTIES OF A RELATION ON SET USING MATRIX :-
There are generally four types of
properties in a set .
i.e. (i)Reflexive
(ii)Symmetric
(iii)Anti-
Symmetric
(iv)Transitive

7. (i) REFLEXIVE BY USING MATRIX :-
For reflexive relation we have (ai,ai)∈R , ⍱ai∈A
⇒Mii=1
i.e. if a matrix have all diagonal is equal to 1 then it
is called reflexive matrix .
So all the diagonals elements are 1 .
i.e. MR=
1 0 0
0
0 1 0
0
0 0 1
0
0 0 0

8. (ii)SYMMETRIC BY USING MATRIX :-
 A relation is said to be a symmetric if
(ai,aj)∈R ⇒ (aj,ai)∈R i.e. Mij=Mij=1, where i
≠j .
 A matrix is said to be symmetric if A=A
T
Ex :- A= 1
0
0
1
A =
T
1 0
0 1

9. (iii)ANTI-SYMMETRICBY USING MATRIX :-
 A relation is anti-symmetric if (ai,aj)∈R , (aj,ai)
∈R ⇒ ai=aj , where i=j .
 We always remember that , in antisymmetric
relation Mij=Mji=0 or Mij=1, Mji=0 or Mij=0 , Mji=1
.
 Ex :- R=
1 0 0 0
0 1 1 0
0 0 0 1
0 0 0 0
 Here M23=1 and M32=0,so it
is a anti-symmetric matrix.

10. (iv)TRANSITIVE BY USING MATRIX :-
A relation is said to be transitive relation
if (ai,aj)∈R , (aj,ak) ∈R ⇒ (aj,ak)∈R
We always remember that Mij=1 , Mik=1 ⇒
Mik=1 .
Ex :- R= 1 0 0 0
0 1 1 1
0 0 0 1
0 0 0 0
 Here M23=1, M34=1 then M24=1 so
it is a transitive relation over
matrix.

11. THANKYOU