Deakin University SIT292 Assignment 1 - Sets, Relations, and Matrices

This document provides a comprehensive solution to SIT292 Assignment 1, focusing on sets, relations, and matrices. The assignment covers several key concepts, including set cardinality, subsets, intersections, and power sets, with detailed explanations and solutions for each problem. The solution demonstrates understanding of reflexive, symmetric, anti-symmetric, and transitive relations, determining equivalence relations, and constructing directed graphs. Furthermore, the assignment includes matrix operations, specifically the calculation of determinants for given matrices. The solution provides step-by-step explanations and calculations for each question, ensuring a clear understanding of the concepts and methods applied. The assignment is a valuable resource for students studying discrete mathematics and preparing for similar assessments. This assignment is a solved homework assignment for the unit SIT292 offered at Deakin University. The document covers the solution to problems on sets, relations and matrices.

Maths
[DATE]

Question 1
(a) This statement is true since the cardinality of the set C is 5 as it contains 5 elements in
total.
(b) This statement is false since there are no common elements between set A and set B.
(c) This statement is true since A containing elements {1,2,3} is a subset of C but is not
equal to C.
(d) This statement is false since the set A i.e. {1,2,3} is not an element in C. C does not
contain the element {1,2,3}.
(e) This statement is true since A containing elements {1,2,3} is a subset of D but is not
equal to C.
(f) This is false since D would not contain an element [3,4,5}. This is because D contains
subsets of set A and potentially cannot contain 4,5 which are not included in set A.
(g) This is false since D will only comprise of all possible subsets of {1,2,3} and hence
cannot contain the element 4 which is contained in set B.
(h) Since {4} is not present in D owing to D comprising of all possible subsets of {1,2,3},
hence the given statement is false.
(i) The given statement is true since D is a power set of A and would contain the set A also
as every set is a subset of itself.
(j) The given statement is true as there is one common element which exists in both sets B
and C which is {2,3}. Hence the intersection of B and C cannot be a null set.
Question 2
(a) The given relation is reflexive since both (a,a) and (b,b) are captured in the given
relationship.
The given relation is symmetric since if a and b are enrolled in the same class, then b and a
are also enrolled in the same class. Hence (a,b) and (b,a) are both captured.
The given relation is not anti-symmetric since the symmetric relationship has already been
established.
The given relation is transitive since if a and b are in the same class and also b and c are in
the same class, it would logically imply that a and c are also in the same class.
The given relationship would be an equivalence relationship since it is reflexive, symmetric
and transitive.
(b) The given relation is not reflexive since a studies more than b would not imply that a
studies more than a or b studies more than b.
The given relation is not symmetric since a studies more than b does not imply than b studies
more than a.
The given relation is anti-symmetric since the symmetric relationship does not uphold as has
been highlighted above.
1

The given relation is transitive since if a studies more than b and b studies more than c, then
logically a studies more than c.
The given relation is not equivalence as for this to happen, it must be reflexive, symmetric
and transitive which is not true for the given relation.
Question 3
(a) Directed graph of the given relation is shown below.
For each of sets in the relationship, the directed graph would indicate an arrow from the first
element to the last element.
For instance (a,a) would be captured by an arrow directed from a to a.
Similarly, (a,b) would be captured by an arrow directed from a to b.
This has been done for all the sets included in the relationship. This had led to the following
figure which reflects all the sets provided in the given relationship.
(b) The given relation is not equivalence. The appropriate reasons are highlighted as follows.
 The relation is not reflexive since elements (a,a) and (b,b) are absent.
 The relation is not symmetric since (a,b) is present but (b,a) is absent. Similarly (c,b)
is present but (b,c) is absent.
2

 The relation is not transitive since (a,c) and (c,a) are present but (a,a) is not present.
Similarly (b,c) and (c,b) is present but (b,b) is absent.
The following elements need to be included so as to make the given relation equivalence.
(a,a) , (b,b) , (b,c)
Question 4
(a) þ 12
Set A = {1,2,3,4}
þ 1= {( 1,2 ) , ( 1,3 ) , ( 2,2 ) , ( 2,3 ) , ( 3,3 ) , ( 4,4 ) }
þ 2= { ( 2,4 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 4,3 ) }
In matrix forms
þ 1=
[0 1 1¿ 0
0 1 1¿ 0
0 0 1 ¿ 0
0 0 0 ¿ 1 ]
þ 2=
[0 0 0 ¿ 0
0 0 0 ¿ 1
1 1 1¿ 0
0 0 1 ¿ 0 ]
Now,
þ 12={ ( 1,2 ) , ( 1,3 ) , ( 2,2 ) , ( 2,3 ) , ( 3,3 ) , ( 4,4 ) }
(b) þ 13 taking n= 3
3

þ 13={ (1,2 ) , ( 1,3 ) , ( 2,2 ) , ( 2,3 ) , ( 3,3 ) , ( 4,4 ) }
(c) þ 1∗þ 2
þ1* þ2={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(4,3)}
(d) þ 2∗þ 1
þ2* þ1={(2,4),(3,2),(3,3),(4,3)}
(e) þ 1∗þ 2∗þ1
þ1*þ2*þ1= {(1,2), (1,3), (1,4), (2,2), (2,3),(2,4),(3,2),(3,3),(4,3)}
Question 5
Determination of matrix A
4

The determination of matrix A comes out to be -72.
Determination of matrix B
5

6

The determination of matrix A comes out to be 0.
7