Relation Matrix & Graphs in Relations

1. Relation Matrix & Graph
Ms. Rachana Pathak
(rachanarpathak@gmail.com)
Assistant Professor, Dept of Computer Science and Engineering
Walchand Institute of Technology, Solapur
(www.witsolapur.org)

2. Learning Outcome
2Walchand Institute of Technology, Solapur
At the end of this session,
Students will be able to evaluate relation matrix and graphs on it.

3. Prerequisite
• Basics of Discrete Mathematics
• Basics of Relation
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4. Introduction
Relation Matrix
• A relation R from a finite set A to a finite set B can be
represented by a matrix called the relation matrix of R.
• Let A ={a1,a2,a3…am} and B= {b1,b2,b3……bn} be finite set
containing m and n elements, respectively, and R be the
relation from A to B.
• Then R can be represented by an m x n matrix Mr = [rij],which
is defined as follows:
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5. rij = 1, if ai R bj
0, if ai R bj
Note that the matrix MR has the elements as 1’s and 0’s.
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6. Example
Let A = {1,2,3,4} and B ={b1,b2,b3}. Consider the relation R =
{(1,b2),(1,b3),(3,b2),(4,b1),(4,b3)}. Determine the matrix of the
relation.
Solution :
A = {1,2,3,4} B = {b1,b2,b3}.
Relation R = {(1,b2),(1,b3),(3,b2),(4,b1),(4,b3)}.
Matrix of the Relation R is written as-
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7. Example
b1 b2 b3
1 0 1 1
2 0 0 0
3 0 1 0
4 1 0 1
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8. Example
0 1 1
0 0 0
0 1 0
1 0 1
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9. Think & Write?
Let A = {1,2,3,4}. Find the relation R on A determined by the
matrix.
MR = 1 0 1 0
0 0 1 0
1 0 0 0
1 1 0 1
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10. Answer
The relation R =
{ (1,1),(1,3),(2,3),(3,1),(4,1),(4,2),(4,4) }
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11. Properties of a Relation in a Set
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• All diagonal entries must
be 1
Reflexive
• Rij = Rij for every i and jSymmetric
• Rij = 1 & Rji = 0 for i≠jAnti-symmetric

12. Reflexive
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MR = 1 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1

13. Symmetric
MR = 1 2 3 4
1
2
3
4
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1 1 0 1
1 1 0 0
0 0 0 0
1 0 0 0

14. Anti-Symmetric
MR = 1 2 3 4
1
2
3
4
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1 0 0 0
1 1 0 0
0 0 0 0
1 0 0 0

15. Graph
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• A relation defined in a finite set can also be represented pictorially with
the help of a graph.
• Let R be a relation in a finite set A = {a1,a2,….an}.
• Elements of A are represented by points or circles called nodes.
• These nodes are called vertices
• Arcs are used to show the connection called as edge.
• Let us see some examples :

16. .b
. a aRb
.
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Here we say a
is in relation
with a
^
Here we say a
is in relation
with b

17. “a is in relation with b and b is in relation
with b”
“ a is in relation with b and b in relation with
c and c is in reltion with a”
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Image source : 1. Discrete Mathematics with combinatorics and graph theory- S. SANTHA (CENGAGE Learning)

18. Conclusion :
In this session, We have studied all about POSET.
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19. References
• 1. Discrete mathematical structures with applications to computer science -- J. P.
Tremblay & R. Manohar (MGH International)
• Reference Books:
• 1. Discrete Mathematics with combinatorics and graph theory- S. SANTHA
(CENGAGE Learning)
• 2. Discrete Mathematical Structures – Bernard Kolman, Robert C. Busby (Pearson
Education)
• 3. Discrete mathematics -- Liu (MGH)
19Walchand Institute of Technology, Solapur

20. Thank You !!!
20Walchand Institute of Technology, Solapur