The document consists of an assignment covering topics such as partitions, equivalence relations, and Hasse diagrams. It includes tasks related to generating equivalence relations, creating relation matrices, and drawing Hasse diagrams for various sets under partial ordering. The assignment aims to explore the properties and interactions of mathematical relations and provide examples for clarification.

1. DMS Sunita M Dol
Walchand Institute of Technology, Solapur Page 1
Assignment No. 10
Topics covered:
• Partition and covering of set
• Equivalence relation
• Composition of binary relations
• POSET and Hasse diagram
1. Given a set S = {1, 2, 3, 4, 5}, find the equivalence relation on S which
generates the partition {{1, 2}, {3}, {4, 5}}. Draw the graph of the relation.
2. Given the relation matrix MR of a relation R on the set {a, b, c}, find the
relation matrices of R~
, R2
= R ◦ R, R3
= R ◦ R ◦ R, and R ◦ R~
MR =
3. Two equivalence relation R and S are given by their relation matrices MR
and MS. Show that R ◦ S is not equivalence relation.
MR= MS=
Obtain equivalence relations R1 and R2 on {1, 2, 3} such that R1 ◦ R2 is
also an equivalence relation.
4. Draw the Hasse diagrams of the following sets under the partial ordering
relation “divides” and indicate those which are totally ordered
a. {2, 6, 24}
b. {3, 5, 15}
c. {1, 2, 3, 6, 12}
d. {2, 4, 8, 16}
e. {3, 9, 27, 54}
1 0 1
1 1 0
1 1 1
1 1 0
1 1 0
0 0 1
1 1 0
1 1 1
0 1 1

2. DMS
Walchand Institute of Technology, Solapur
5. Figure gives the Hasse diagram of a partially ordered set <P, R> where P =
{x1, x2, x3, x4, x5}
Find which of the following are true
a. x1Rx2
b. x4Rx1
c. x3Rx5
d. x2Rx5
e. x1Rx1
f. x2Rx3
g. x4Rx5
Walchand Institute of Technology, Solapur
Hasse diagram of a partially ordered set <P, R> where P =
{x1, x2, x3, x4, x5}
Find which of the following are true
Sunita M Dol
Page 2
Hasse diagram of a partially ordered set <P, R> where P =