SIT192 Discrete Mathematics: Relations, Graphs, and Properties

This document presents a solution to a discrete mathematics assignment from Deakin College, Trimester 1, 2019. The assignment covers topics such as relations, their properties (reflexive, symmetric, transitive, antisymmetric), matrix representations of relations, and directed graphs. Several questions are addressed, including determining whether given relations satisfy specific properties, finding matrix representations and drawing directed graphs for given relations on the set A = {1, 3, 5, 7, 9}, and analyzing a relation defined by modular arithmetic. The solution provides detailed explanations and justifications for each answer, along with visual representations of graphs and matrices. The assignment also includes a question on adjacency matrices representing draws between teams and the properties of the matrix. The document concludes with a directed graph representing the draw.

ASSIGNMENT ON DISCRETE MATHEMATICS 1
ASSIGNMENT ON DISCRETE MATHEMATICS
By (Firstname Lastname)
Course Name
Professor (Tutor)
Deakin College
May 30, 2019

ASSIGNMENT ON DISCRETE MATHEMATICS 2
Question 2
The matrix representation for the part (i) of the equation i.e
R= { ( 1 ,9 ) , ( 3 , 7 ) , ( 5 ,5 ) , ( 7 , 3 ) , ( 9 ,1 ) , ( 5 ,7 ) ,(7,9) }
The matrix reads
The graph for the above matrix is drawn as shown below
Question 2 (ii)
For this section, we know that,
since 3−1
2 =1∈ Α ,therefore ( 3,1 ) ∈ R ,∧since 7−1
2 =3 ∈ A , therefore , ( 7,1 ) ∈ R

ASSIGNMENT ON DISCRETE MATHEMATICS 3
ince 5−3
2 =1 ∈ Α , therefore ( 5,3 ) ∈ R ,∧since 9−3
2 =3 ∈ A , therefore , ( 9,3 ) ∈ R
ince 7−5
2 =1 ∈ Α , therefore ( 7,5 ) ∈ R ,∧since 9−7
2 =1∈ A , therefore , ( 9,7 ) ∈ R
R= { ( 3,1 ) , ( 7 , 1 ) , ( 5 , 3 ) , ( 9 , 3 ) , ( 7 ,5 ) , ( 9 , 7 ) }
The matrix reads
Question 2 (iii)
For this section we have, R= { ( a , b ) :a+b≡ 1(mod 5) }
Solution
if 1+5 ≡1 ( mod 5 ) . therefore , ( 1,5 ) ∈ R
if 3+3 ≡ 1 ( mod 5 ) . therefore , ( 3,3 ) ∈ R
if 5+1 ≡1 ( mod 5 ) . therefore , ( 5,1 ) ∈ R
if 7+ 9≡ 1 ( mod 5 ) . therefore , ( 7,9 ) ∈ R
if 9+ 7≡ 1 ( mod 5 ) . therefore , ( 9,7 ) ∈ R

ASSIGNMENT ON DISCRETE MATHEMATICS 4
Therefore, our matrix representation for R= { ( 1,5 ) , ( 3 , 3 ) , ( 5 ,1 ) , ( 7 , 9 ) , ( 9 ,7 ) } becomes
The directed graph is drawn as shown below

ASSIGNMENT ON DISCRETE MATHEMATICS 5
Question 3: Whether R satisfies the following properties
d) Antisymmetric since for every (a; b), we do not have (b, a ) .

ASSIGNMENT ON DISCRETE MATHEMATICS 6
d) Not antisymmetric since we have both (1; 9) and (9; 1), ( 7, 9 ) and ( 9, 7)
d) Not antisymmetric as we have both (5; 9) and (9; 9)
Question 5 part (i)
an adjacency matrix representing the draw
A B C D E F
A * 1 0 1 0 1
B 0 * 1 0 1 0
C 1 0 * 1 0 1
D 0 1 0 * 1 0
E 1 0 1 0 * 1
F 0 1 0 1 0 *

ASSIGNMENT ON DISCRETE MATHEMATICS 7
The matrix above we take the first team as the home team.
Since for instance the relation between A versus B, ( A, B) is 1 but instead ( B, A) is zero, this
indicates that the matrix is not symmetric.
Question 5 (ii)
Directed graph representing the draw
Consider the graph below