Assignment 4
COEN 231: Introduction to Discrete Mathematics
Due Date: 4 Dec 2023
Problem1
Consider the following relations on the set of real numbers:
Problem2
Write all equivalence class of Congruence module 3.
Problem3
Let R be the relation on the set of all people who have visited a particular Web page
such that x Ry if and only if person x and person y have followed the same set of links starting at this
Web page (going from Web page to Web page until they stop using the Web). Show that R is an
equivalence relation.
Problem4
Determine whether the relations represented by these zero–one matrices are
equivalence relations.
a)
b)
c)
Problem5
Show that the relation R on the set of all bit strings such that s R t if and only if s and t
contain the same number of 1s is an equivalence relation. What is the equivalence class of the bit
string 011 for R?
Problem6
The word apple can refer to a plant, a food, or a computer company. Construct a word
graph for these nouns: apple, strawberry, lenovo, cheese, chocolate, ibm, oak, microsoft, hedge,
grass, cake, quiche, hp, cider, donut, azalea, pine, dell, fir, raspberry. Connect two vertices by an
undirected edge if the nouns they represent have similar meaning.
Problem7
Which of the following relations on {0, 1, 2, 3} are partial orderings? Determine the
properties of a partial ordering that the others lack.
a) {(0, 0), (2, 2), (3, 3)}
b) {(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 3)}
c) {(0, 0), (1, 1), (1, 2), (2, 2), (3, 1), (3, 3)}
d) {(0, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (2, 3), (3, 0), (3, 3)}
e) {(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 3)}
Assignment 4
COEN 231: Introduction to Discrete Mathematics
Due Date: 4 Dec 2023
Problem8
For each of the following graph determine the type of the graphs: Directed, Not
directed, simple, multigraph, Pseudograph. For the one that is not simple, find a set of edges to
remove to make it simple.
G1
G4
G2
G5
G3
G6
Problem9
Use paths either to show that these graphs are not isomorphic or to find an
isomorphism between these graphs.
Problem10
In the following graph check whether the given graph has an
a. Euler circuit
b. Euler path
c. Hamilton Circuit
d. Hamilton path
Problem11
Use Prim’s algorithm to find a minimum spanning tree for the given
weighted graph.