The document outlines an assignment covering topics related to relations, including their properties and representations. It includes several problems involving union, intersection, range, and specific examples of relations such as transitive, symmetric, and antisymmetric. Additionally, it presents tasks to analyze properties of defined relations on sets and verify certain characteristics.

1. DMS Sunita M Dol
Walchand Institute of Technology, Solapur Page 1
Assignment No. 9
Topics covered:
• Relations
• Properties of binary relations
• Matrix and graph representation
1. Let P = {<1, 2>, <2, 4>, <3, 3>} and Q = {<1, 3>, <2, 4>, <4, 2>}. Find
a. P ∪ Q,
b. P ∩ Q,
c. D(P),
d. D(Q),
e. D(P ∪ Q),
f. R(P),
g. R(Q) and
h. R(P ∩ Q).
Show that
D(P ∪ Q) = D(P) ∪ D(Q) and
R(P ∩ Q) ⊆ R(P) ∩ R (Q)
2. What are the ranges of the relations
S = {<x, x2> | x ∈ N}
R = {<x, 2x> | x ∈ N}
where N = {0, 1, 2, 3, …}? Find R ∪ S and R ∩ S
3. Let L denote the relation “less than or equal to” and D denote the relation
“divides” where xDy means “x divides y”. Both L and D are defined on the
set {1, 2, 3, 6}. Write L and D sets and find L ∩ D.
4. Give an example of a relation which is neither reflexive nor Irreflexive.
5. Give an example of a relation which is both symmetric and Antisymmetric.

2. DMS Sunita M Dol
Walchand Institute of Technology, Solapur Page 2
6. Show that whether the following relations are transitive
R1 = {<1, 1>}
R2 = {<1, 2>, <2, 2>}
R3 = {<1, 2>, <2, 3>, <1, 3>, <2, 1>}
7. Given S = {1, 2, 3, 4} and a relation R on S defined by
R = {<1, 2>, <4, 3>, <2, 2>, <2, 1>, <3, 1>}
Show that R is not transitive. Find a relation R1 ⊇ R such that R1 is
transitive. Can you find another relation R2 ⊇ R which is also transitive?
8. Given S = {1, 2, …, 10} and a relation R on S where
R = {<x, y> | x + y = 10}
What are the properties of the relation R?