Experience Teaching Mathematics at the University of Defence in the
Study Field of Cybersecurity
POTŮČEK R.
Department of Mathematics and Physics, University of Defence,
Kounicova 65, 662 10 Brno,
e CZECH REPUBLIC
hORCID iD: 0000-0003-4385-691X
Abstract: The teaching of mathematics at the Faculty of Military Technology of the University of Defence in
the first two years of the five-year Military master’s degree is divided into the teaching of students in four
specializations: Military Technologies – Mechanical, Military Technologies – Electrical, Military Geography and
Cartography, and Cybersecurity. The article discusses the previous five-year experience in teaching the subjects
Mathematics I, Mathematics II, Mathematics III, and Graph theory for the specialization in Cybersecurity, that the
author of the paper gained while teaching, as a guarantor of these four subjects. The article provides an overview
of the topics that are covered, typical exam assignments, and an evaluation of the results of the semester exams in
the five years since the accreditation of the Cybersecurity specialization was granted.
Key-Words: University of Defence, Faculty of Military Technology, Cybersecurity study program, teaching
mathematics, teaching graph theory, semester exam, classified credit test, thematic units, exam evaluation
Received: March 13, 2024. Revised: October 6, 2024. Accepted: November 3, 2024. Published: December 2, 2024.
1 Introduction
This paper deals with the problems of teaching
mathematics, and especially discrete mathematics, at
the Faculty of Military Technology at the University
of Defence in Brno. The tradition of teaching
mathematics at this high military school has its
origins in 1951, when this school was established.
We will deal with mathematical education and the
ranges and thematic areas of mathematics teaching
in the Cybersecurity study program, which has been
accredited at the Faculty of Military Technology since
2019. The paper, [1], describes the Cybersecurity
study program from the perspective of teaching
mathematics. New trends in the education of
mathematics around the world can be found, for
example, in the paper, [2].
Mathematics teaching in the study field of
Cybersecurity takes place during the first four
semesters of this five-year master’s degree program.
In the first, second, and third semesters, mathematics
teaching consists of lectures and exercises in the
subjects of Mathematics I, Mathematics II, and
Mathematics III. Two mathematical subjects are
taught in the fourth semester: Graph theory and
Probability theory along with mathematical statistics.
Since the author of the paper does not teach the
subject of Probability and mathematical statistics, this
paper will focus on the content, topics, semester
examinations, and the success of students in the
subjects of Mathematics I, II, and III, as well as
the subject of Graph theory. Detailed information
about the accredited study program Cybersecurity
can be found on the website, [3]. Papers dealing
with mathematics teaching of Cybersecurity study
program at foreign universities include, for example,
[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14],
[15], [16], [17]. On the Internet, the relationship
between mathematics and cybersecurity is engaged
in, for example, articles, [18], [19], [20], [21], [22],
[23], [24], [25], [26], [27], [28].
In contrast, the study of mathematics in the Master
study program Military Technologies – Mechanical
and Military Technologies – Electrical includes
this thematic units: Linear algebra and analytic
geometry, Real functions and differential calculus
of one variable, Integral calculus of one variable,
Differential calculus of several variables, Ordinary
differential equations, Infinite and Fourier series,
Probability and mathematical statistics, Multiple
integrals, Vector analysis, line and surface integrals,
Complex analysis, and Numerical methods of algebra
and analysis.
Between the academic years 2019/2020 to
2022/2023, civil students were also admitted to
the study of Cybersecurity, in addition to military
students and students from the Socialist Republic of
Vietnam. Since the academic year 2023/2024, only
students of military studies have been accepted. The
number of students who take part in the entrance
exams for the Cybersecurity field of study is usually
between 50 and 80 applicants. Information on the
entrance tests at the Faculty of Military Technology
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can be found in the paper [29]. The number of
students admitted to study ranges from 20 to 40
students, that form one or two learning groups.
2 Mathematics I
The teaching of this subject is subsidized for
90 lessons. The lecture and exercises each
have a duration of two lessons, so it consists
of 45 learning units. The teaching consists of
22 lectures and 23 exercises, including 2 laboratory
exercises being conducted in a computer classroom.
The mathematical software used in the laboratory
exercises in Mathematics I, II, III and Graph theory
is computer algebra system Maple.
2.1 Topics of Mathematics I
Individual topics discussed within the subject
Mathematics I are stated in the following Table 1,
which, in addition to the topics, includes hourly
subsidies (HS) for lectures (L), exercises (E) and
laboratory exercises in a computer classroom (C).
Table 1.Topics of Mathematics I with their hourly
subsidies (L/E/C)
#Topic HS L/E/C
1 Mathematical logic 8 4/4/0
2 Set theory 8 4/4/0
3 Binary relations 8 4/2/2
4 Mapping 4 2/2/0
5 Partially ordered sets 4 2/2/0
6 Combinatorics 8 4/4/0
7 Fields nad vector spaces 8 4/4/0
8 Matrices and matrix operations 8 4/4/0
9 Solving systems of linear equations 10 4/4/2
10 Vector subspaces and linear span 4 2/2/0
11 Intersection and sum of vector subspaces 4 2/2/0
12 Basis and dimension of vector space 8 4/4/0
13 Linear mapping 8 4/4/0
2.2 Semester Exam in Mathematics I
Students are always introduced to the content of the
semester exam at the beginning of the semester. At
the end of the semester, several sets (usually four)
of solved preparatory tasks for the exam are made
available to students for the purposes of independent
study. The semester exam in Mathematics I consist
of a theoretical part and a practical part.
The theoretical part contains five questions about
theory. From 0 to 4 points can be obtained for
each answered question, so a student can receive a
maximum of 20 points for the theoretical part. The
content of the theoretical questions are the definitions
of the most important terms, their basic properties
and the most important sentences of the relevant
theoretical unit. Part of the theoretical question can
also be made up of a simple example illustrating the
given concept. A maximum of another 20 points,
which are included in the total point gain in the exam,
can be gained by the student from exercises based on
partial written works and homework.
The practical part of the exam consists of the
assignment of six tasks. These tasks evenly cover the
topics presented at the lectures and correspond to the
assignment of solved preparatory examples. For each
task, the student can get from 0 to 10 points, so for
the practical part, the student can receive a maximum
of 60 points. For the semester exam, the student can
get a maximum of 100 points, while at least 50 points
must be achieved to successfully pass the exam.
2.3 Example of a Semester Exam in
Mathematics I
One of the assignments of the semester exam had
the following theoretical and practical part. The
theoretical part lasts 30 minutes, and after a short
break, the practical part follows, for which 120
minutes are reserved.
▶..............Thetheoreticalpart ..............
A. Define the power set of the n-element set Mand
its cardinality.
Determine the power set of the 3-element set M=
{a, b, c}and its cardinality. [4 points]
B. Define the RS composition of the binary relations
R⊂A×BaS⊂B×C.
Write a relation for (RS)−1for binary relations
R⊂A×Band S⊂B×C. [4 points]
C. Define the least element and the minimal element
of the ordered set (A, ≤).
What is the relationship between the least and the
minimal element of an ordered set (A, ≤)?
[4 points]
D. Write a necessary and sufficient condition for the
algebraic structure (T, +,·)to be a field.
Give an example of an infinite field and an
example of a finite field. [4 points]
E. Define nlinearly dependent vectors and nlinearly
independent vectors. [4 points]
▶...............Thepracticalpart ...............
1) Verify that the propositional formula
(p∧q)↔r→p
is consistent, express it in the conjunctive normal
form and simplify as much as possible by using
the algebraic properties of logical connectives.
[10 points]
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2) Determine how many different 4-digit numbers
that are divisible by 5 can be formed from the six
digits 0,1,3,5,7,9if the digits
a) cannot repeat, b) can repeat? [10 points]
3) On the set A={1,4,9,12,17,20}, the relation R
is defined such that
(x, y)∈R↔x−y= 8kfor some k∈Z.
Determine the number of elements of the relation
R(i.e. the number of ordered pairs (x, y)∈A2
that satisfy the relation R).
Prove that Ris an equivalence relation.
Determine the partition of the set Ainto the
equivalence classes A/R. [10 points]
4) Decide whether the basis of the vector space of the
matrices R2×2can be formed by the matrices
A=0 1
2 3, B =1 2
3 0, C =2 3
0 1,
D=1 2
3 4.[10 points]
5) Determine some basis of the subspace Uof
the vector space P2generated by the vectors
(polynomials) P(x) = x2−x+ 2,Q(x) =
x2+x−1,R(x) = x2−3x+ 5.
Fill the basis of the subspace Uwith suitable
vectors from the canonical basis of a vector space
of polynomials of degree at most 2 on the basis B
of the vector space P2.
Determine the coordinates of the vector T(x) = x
with respect to the chosen basis B. [10 points]
6) Determine the matrix of linear mapping
L:R3→R2given by the relation
L(x, y, z) = (x+ 2y, y −2z)
with respect to the bases G=(1,1,0),(1,0,1),
(0,1,1)and H=(1,1),(0,1)of the vector
spaces R3and R2.
Determine the image of the vector xin the linear
mapping Lif xG= (1,0,−1)T. [10 points]
2.4 Students’ Success in the Mathematics I
Exam
The following Table 2 presents the success of
students in the semester exam in Mathematics I
in the winter semesters of the academic years
(AY) 2019/2020, 2020/2021, 2021/2022, 2022/2023,
2023/2024, cumulatively for the academic years
2019/2020 to 2023/2024 and also the total number
(TN) of students who continue to study successfully
and did not leave their studies or left because of failure
to fulfill their study obligations.
Table 2.Final evaluations for the semester exam
in Mathematics I in the academic years 2019/2020,
2020/2021, 2021/2022, 2022/2023, 2023/2024 and
from the academic year 2019/2020 to 2023/2024
AY A B C D E FE FFE TN
2019/2020 3 3 2 1 4 3 1 17
2020/2021 3 1 4 4 4 1 1 18
2021/2022 4 6 6 11 3 0 0 30
2022/2023 0 1 2 4 3 3 1 14
2023/2024 1 0 4 5 6 6 4 26
19/20–23/24 11 11 18 25 20 13 7 105
Figure 1 shows the percentage success rate of
students in semester exams in Mathematics I from the
academic year 2019/2020 to 2023/2024, with the fact
that grade E also includes grade E in the 1st or 2nd
correction period. If the student rejected grades B, C
or D and wrote a repair examination, only the final
grade is written in this table.
Fig.1: The percentage success rate in the
Mathematics I exam from AY 2019/20 to 2023/24.
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2.5 Students’ Success in Solving Single Types
of Tasks in the Mathematics I Exam
Based on experience with semester exams, the author
can state that the success of students in solving
individual types of tasks in semester exams does
not change significantly in individual years. The
following Table 3, therefore, contains the evaluation
of success in the exam in AY 2023/2024.
Table 3.Success of students in the task solving at
exam in Mathematics I in AY 2023/2024
#Type of exam task NS AS
1 Checking whether a propositional
formula is a tautology / contradiction
/ consistent, conversion to disjunctive
normal form.
20 6.95
2 Proof of the equality of expressions
with natural numbers by mathematical
induction. // Proof of the statement of
divisibility by mathematical induction.
29 4.59
3 Combinatorics – combinatorial principles
(product, sum, inclusion and exclusion),
permutation, variation, combination
without repetition/with repetition.
49 5.14
4 Binary relations and their properties. //
Verification that the given relation is an
equivalence relation, decomposition into
equivalence classes.
49 3.67
5 Basic operations with matrices. //
Determination the rank of a matrix
depending on a parameter.
22 6.23
6 Verification that vectors / polynomials /
matrices form a basis of the vector space.
// Coordinates of a vector / polynomial /
matrix with respect to the basis.
34 4.47
7 Determination the dimension and
basis of a vector subspace and its
completion based on the vector space. //
Determination the basis and dimension of
the intersection and the sum of two vector
subspaces of the given vector space.
49 4.16
8 Determination the kernel, image,
defect, and rank of a linear map. //
Verification of the linearity of the map
and determination of its matrix.
42 3.17
Traditionally, solving task concerning linear map
and task to verify whether a given relation is an
equivalence relation have the least success rate. On
the contrary, students achieve the best results when
solving a task from propositional logic and a task
concerning matrix operations.
At the beginning of the semester, students are
acquainted with the eight types of tasks that can be
expected in the semester exam. These types of tasks
are listed in the second column of Table 3. The third
column contains the number of students (NS) who
solved this type of task. The fourth column always
shows the average score (AS), i.e. the number of
points always out of the maximum number of 10. The
average score received by students on the theoretical
part of the exam on a total of 49 exams was only 4.86
out of 20 points.
3 Mathematics II
The teaching of the subject Mathematics II is
subsidized for 90 lessons. 45 learning units consist of
22 lectures and 23 exercises, including 3 laboratory
exercises.
3.1 Topics of Mathematics II
Individual topics of subject Mathematics II are stated
in the following Table 4.
Table 4.Topics of Mathematics II with their hourly
subsidies
#Topic HS L/E/C
1 Determinants, their calculation and
application
8 4/4/0
2 Inverse matrices, their calculation and
applications
4 2/2/0
3 Linear transformation, transition matrix 8 4/4/0
4 Eigenvalues and eigenvectors of a
matrix and their calculation
8 4/2/2
5 Similar matrices, diagonalization of
matrices, matrix functions
4 2/2/0
6 Scalar product, Euclidean space,
orthogonal vectors
8 4/4/0
7 Finite and iterative methods of solving
systems of linear equations
8 4/2/2
8 Divisibility criteria, Euclidean
algorithm, Bézout’s identity
4 2/2/0
9 Prime numbers and composite numbers,
Euclid’s theorem
4 2/2/0
10 Congruence relation, linear congruence
equations and their systems
6 2/4/0
11 Euler’s totient function, Fermat’s little
theorem, Euler’s theorem
4 2/0/2
12 Groupoids, semigroups, groups 8 4/4/0
13 Subgroups, cyclic groups, factor groups 8 4/4/0
14 Rings, polynomial rings, Galois fields,
lattices, Boolean algebras
8 4/4/0
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3.2 Semester Exam in Mathematics II
Students are again, just as in Mathematics I,
introduced to the content of the semester exam at the
beginning of the semester. At the end of the semester,
several sets (usually from 4 to 6) of solved preparatory
tasks for the exam are made available to students.
The semester exam in Mathematics II consist of a
theoretical part and a practical part.
The theoretical part contains five questions about
theory. From 0 to 4 points can be obtained for
each answered question, so a student can receive
a maximum of 20 points for the theoretical part.
A maximum of another 20 points, which are included
in the total point gain in the exam, can be gained by
the student from exercises.
The practical part of the exam consists of the
assignment of six tasks. For each task, the student
can get from 0 to 10 points, so for the practical part,
the student can receive a maximum of 60 points. For
the semester exam, the student can get a maximum of
100 points.
3.3 Example of a Semester Exam in
Mathematics II
One of the assignments of the semester exam had
the following theoretical and practical part. The
theoretical part lasts 30 minutes, the practical part
follows, for which 120 minutes are reserved. Thus,
the Mathematics II exam took 150 minutes, as was
the case with the Mathematics I exam.
▶.............Thetheoreticalpart .............
A. Formulate Cramer’s rule, give two applications
of determinants in linear algebra or in analytical
geometry. [4 points]
B. Write the axiomatic definition of the scalar
product of the vectors. [4 points]
C. Convert a system of linear equations A·x=
=bto the iteration form expressing (k+1)-th
approximation its solution x(k+1). [4 points]
D. Define a monoid, give two examples of monoids,
formulate three inversion properties in the
monoid. [4 points]
E. Define a cyclic group and a group generator, give
two examples of the cyclic group. [4 points]
▶..............Thepracticalpart ..............
1) Solve the matrix equation A·X·B=C,
where A=2 1
3 2 ,B=110
131
011,C=
3 8 3
5 13 5 . [10 points]
2) Verify that the vectors u1= (1,1,1),u2=
(1,1,0),u3= (1,0,0) form the basis of the
Euclidean vector space R3with a standard scalar
product, and use the Gram–Schmidt process to
determine the orthonormal basis. [10 points]
3) Using matrix diagonalization (not matrix
multiplication), calculate the 8th power A8of
the matrix A=0 1
−2 3 . [10 points]
4) Using modular arithmetic (not just using calculator)
for given numbers a= 1311,b= 9:
a) determine the rest after dividing aby b,b)
determine the last digit of a,c) show that the
number (a·b+ 17) is divisible by 50. [10 points]
5) There are given two algebraic structures Z4,⊕,
where x⊕y= (x+y)(mod 4) and Z∗
8,⊙=
= ({1,3,5,7},⊙), where a⊙b= (a·b)(mod 8).
a) Create Cayley tables of both algebraic
structures. b) Verify, that both structures form a
group. c) Determine all subgrups of both groups.
d) Decide whether these groups are isomorphic,
and justify your claim. [10 points]
6) In a set of all polynomials Z7[x]write down the
Bézout’s identity for polynomials P(x) = 6x3+
5x2+ 4x+ 3 and Q(x) = 2x+ 1. [10 points]
3.4 Students’ Success in the Mathematics II
Exam
The following Table 5 presents the success of
students in the semester exam in Mathematics II
in the summer semesters of the academic years
2019/2020, 2020/2021, 2021/2022, 2022/2023,
2023/2024 cumulatively for the academic years
2019/2020 to 2023/2024 and the total number of
students who continue to study successfully.
Table 5.Final evaluations for the semester exam
in Mathematics II in the academic years 2019/2020,
2020/2021, 2021/2022, 2022/2023, 2023/2024 and
from the academic year 2019/2020 to 2023/2024
AY A B C D E FE FFE TN
2019/2020 7 3 1 1 2 2 0 16
2020/2021 4 3 4 4 1 2 1 19
2021/2022 4 8 10 5 3 0 0 30
2022/2023 1 2 1 4 3 2 0 13
2023/2024 1 3 4 4 8 2 2 24
19/20–23/24 17 19 20 18 17 8 3 102
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Figure 2 shows the percentage success rate of
students in semester exams in Mathematics II from
the academic year 2019/2020 to 2023/2024, with the
fact that grade E also includes grade E in the 1st or
2nd correction period. If the student rejected grades
B, C or D and wrote a repair examination, only the
final grade is written in this table.
Fig.2: The percentage success rate in the
Mathematics II exam from AY 2019/20 to 2023/24
3.5 Students’ Success in Solving Single Types
of Tasks in the Mathematics II Exam
The following Table 6 contains the evaluation of
success in the last exam in AY 2023/2024.
The task concerning polynomial factor rings and
the task to determine the orthogonal basis along
with the task concerning geometrical applications
of different types of vector products have the least
success rate. The highest scores achieve a task
concerning isomorphic algebraic structures and their
properties and the task to determine the greatest
common divisor (GCD) and the least common
multiple (LCM) of two integers along with the task
of solving a system of linear congruence equations.
At the beginning of the semester, students are
acquainted with the eight types of tasks that are stated
in the second column of Table 6. The third column
contains the number of students who solved this type
of task. The fourth column always shows the average
score, i.e. the number of points that students achieved
from a maximum of 10 points. The average score
received by students on the theoretical part of the
exam on a total of 27 exams was 6.85 out of 20 points.
Table 6.Success of students in the task solving at
exam in Mathematics II in AY 2023/2024
#Type of exam task NS AS
1 Solving a matrix equation of type
AX =Bor XA =Bor AXB =C
using an inverse matrix.
23 6.35
2 Determination the transition matrix
from basis to basis, determination of the
coordinates of the vector on a given basis.
7 6.57
3 Determination the orthogonal and
orthonormal bases of the subspace of
Euclidean space products. // Geometric
applications of scalar, vector and triple
products.
27 4.33
4 Determination eigenvalues and
eigenvectors of a given matrix. //
Calculation of the power of the matrix
using its diagonalization.
27 4.56
5 Determination GCD and LCM of two
integers, Euclidean algorithm, Bézout’s
identity. // Solving a system of linear
congruence equations by an elimination
method.
24 6.75
6 Solving the linear congruence equation
using Euler’s theorem. // Determination
the remainder after dividing two natural
numbers, determination the latest digits of
the natural number in the form of a power.
14 6.00
7 Creating Cayley tables of two algebraic
structures, verification of the properties
of the group, determination its subgroups
and the justification of whether these
groups are isomorphic or determination
group generators.
27 7.52
8 Determination GCD and Bézout’s
identity for polynomials in Zp[x]. //
Solving the linear polynomial equation
in the factor ring.
16 2.31
4 Mathematics III
The teaching of the subject Mathematics III is again,
just as Mathematics I and Mathematics II, subsidized
for 90 lessons. 45 learning units consist of 22 lectures
and 23 exercises, including 2 laboratory exercises.
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4.1 Topics of Mathematics III
The list of lecture topics on the subject Mathematics
III is given by the following Table 7.
Table 7.Topics of Mathematics III with their hourly
subsidies
#Topic HS L/E/C
1 Propositonal logic, disjunctive and
conjunctive normal form
8 4/4/0
2 Algebraic minimization of normal
forms, the Karnaugh maps
8 4/4/0
3 Deductive system, deduction,
correctness and completeness theorems
4 2/2/0
4 Predicate logic 6 2/4/0
5 Proofs in predicate logic, model theory 4 2/2/0
6 Finite-state machines, languages
recognized by the finite-state machine
8 4/4/0
7 Real function of one real variable,
elementary functions
8 4/4/0
8 Limit of a function, continuous
functions
4 2/2/0
9 Derivative of a function, L’Hôspital’s
rule
8 4/4/2
10 Applications of differential calculus,
Taylor polynomials
8 4/4/0
11 Indefinite integral, integration of some
special functions
8 4/4/0
12 Definite integral and its computation 4 2/2/0
13 Improper integral and its computattion 4 2/0/2
14 Sequences, infinite series 4 2/2/0
15 Power series, generating functions 4 2/2/0
4.2 Semester exam in Mathematics III
Students are again always introduced to the content of
the semester exam at the beginning of the semester.
At the end of the semester, several sets of solved
preparatory tasks for the exam are made available
to students. The semester exam in Mathematics III
consist of a theoretical part and a practical part.
The theoretical part contains five questions about
theory. From 0 to 4 points can be obtained for
each answered question, so a student can receive
a maximum of 20 points for the theoretical part.
A maximum of another 20 points can be gained by
the student from exercises.
The practical part of the exam consists of the
assignment of 6 tasks. For each task, the student can
get 0 to 10 points, so for the practical part, the student
can receive a maximum of 60 points. For the semester
exam, the student can get a maximum of 100 points.
4.3 Example of a Semester Exam in
Mathematics III
One of the assignments of the semester exam had
the following theoretical and practical part. The
theoretical part lasts 30 minutes, the practical part
lasts 120 minutes, so the Mathematics III exam took
150 minutes, as the Mathematics I and II exams.
▶.............Thetheoreticalpart .............
A. Define Peirce logical NOR operator and write its
truth table. [4 points]
B. Present the symbols and their names that form the
alphabet of predicate logic. [4 points]
C. Define the root of the polynomial and state how
to determine the integer and rational roots of the
polynomial. [4 points]
D. Define a continuous function at a point, give
the types of discontinuation and state relationship
between continuity and derivatives at that point.
[4 points]
E. Formulate the integral test of the convergence of
an infinite series. [4 points]
▶..............Thepracticalpart ..............
1) Reformulate the sentences into the propositional
formula and decide whether the formula under
the line is a logical consequence of the formulas
above the line: [10 points]
Pavel has a ticket purchased, but urban transport
does not work.
If urban transport works, Pavel will come to work
on time.
Pavel has a ticket purchased and will not come
to work on time.
2) Using an algebraic minimization for the
propositional formula f(p, q, r), which has a
vector evaluation of h= (0,0,0,1,0,0,1,1)
for the ordered triplet of truth values
(0,0,0),(0,0,1), . . . , (1,1,1) of propositional
variables (p, q, r), determine its minimal
disjunctive form. [10 points]
3) Write the state-transition table and decide which
of the words dad,sad,asad,dasa,sada,aadss
the finite-state machine Arecognizes, if Ahas a
set of states q={1,2,3}, alphabet Σ = {a, d, s}
and the state diagram
[10 points]
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4) Function is given by the formula
f:y=π
2+tan2x+π
3.
Determine the inverse function f−1(x)and
a domain D(f)so that for f(x)the function
f−1(x)can exist.
Determine the domain D(f−1)and a range
H(f−1)and draw the graphs of the functions f
and f−1. [10 points]
5) Function is given by the formula
f:y= (1 −2x)e2x−1.
Determine its local minima and maxima, intervals
of monotony and inflexion points. [10 points]
6) Using a definite integral, calculate the volume V
of the revolution solid that is created by the
rotation of the subgraph of the logarithmic
function y=ln x,x∈ ⟨1,e⟩, around the x-axis.
[10 points]
4.4 Students’ Success in the Mathematics III
Exam
The following Table 8 presents the success of students
in the semester exam in Mathematics III in the
winter semesters of the academic years 2020/2021,
2021/2022, 2022/2023, 2023/2024, cumulatively for
the academic years 2020/2021 to 2023/2024 and
the total number of students who continue to study
successfully.
Table 8.Final evaluations for the semester exam in
Mathematics III in the academic years 2020/2021,
2021/2022, 2022/2023, 2023/2024 and from the
academic year 2020/2021 to 2023/2024
AY A B C D E FE FFE TN
2020/2021 4 2 5 0 0 5 0 16
2021/2022 3 3 5 2 1 2 3 19
2022/2023 9 6 4 6 3 2 0 30
2023/2024 0 0 3 4 5 1 0 13
20/21–23/24 16 11 17 12 9 10 3 78
Figure 3 shows the percentage success rate of
students in semester exams in Mathematics III from
the academic year 2020/2021 to 2023/2024, with the
fact that grade E also includes grade E in the 1st or
2nd correction period. If the student rejected grades
B, C or D and wrote a repair examination, only the
final grade is written in this table.
Fig.3: The percentage success rate in the
Mathematics III exam from AY 2020/21 to 2023/24
4.5 Students’ Success in Solving Single Types
of Tasks in the Mathematics III Exam
The Table 9 on the next page contains the evaluation
of success in the last exam in AY 2023/2024.
Students’ success in the semester exam in
Mathematics III was relatively high for the tasks of
the propositional and predicate logic and theory of
finite-state machines, and quite low in the tasks of
the theory of real variable, differential and integral
calculus and of convergence criteria of series.
At the beginning of the semester, students are
acquainted with the eight types of tasks that are stated
in the second column of Table 9. The third column
contains the number of students who solved this type
of task. The fourth column always shows the average
score, i.e. the number of points that students achieved
from a maximum of 10 points. The average score
received by students on the theoretical part of the
exam on a total of 15 exams was 5.47 out of 20 points.
5 Graph Theory
Teaching Graph theory at the Faculty of Military
Technology began in the mid 1990s, when the subject
Graphic algorithms was taught by the author and at
first also one of his colleagues, as the facultative
subject of the doctoral study.
Teaching Graph theory in the master’s degree
for students of the specialization in Cybersecurity,
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Table 9.Success of students in the task solving at
exam in Mathematics III in AY 2023/2024
#Type of exam task NS AS
1 Verification of logical equivalence/logical
consequence by using a truth table. //
Reformulating sentences into logical
formulas and decisions whether a given
logical formula is their logical consequence.
15 9.67
2 Determination of minimal disjunctive forms
of 3 or 4 propositional variables using
algebraic minimization / Karnaugh map.
15 7.40
3 Formal notation of the predicate formula, its
negation and verbal expression of negation.
7 8.14
4 Determination the state-transition table of
the finite-state machine and words that it
recognizes.
8 9.88
5 Determination the inverse function, its
domain and range and drawing graphs of
the function and inverse function.
8 5.50
6 Determination local minima and maxima,
intervals of monotony and inflexion points
of a given function.
14 3.71
7 Evaluation the definite integral by a suitable
integration method. // Calculation the
indefinite integral using partial fraction
decomposition. // Geometric applications
of a definite integral.
15 3.27
8 Decision on convergence of two given series
using a suitable convergence criterion.
8 4.00
which covers the basic thematic units from this
part of discrete mathematics along with the
most important graph algorithms, began until
academic year 2019/2020 after accreditation of the
Cybersecurity study program.
The subject Graph theory, unlike the three
previous subjects, Mathematics I, II and III, does not
end with a semester exam, but a classified credit. The
teaching of the subject Graph theory is subsidized for
60 lessons. 30 learning units consist of 15 lectures
and 15 exercises, including 3 laboratory exercises.
Laboratory exercises from Graph theory, as well
as laboratory exercises in the subjects Mathematics I,
Mathematics II and Mathematics III, take place in
computer classrooms using the computer algebra
system Maple, which, together with the programming
language Matlab, is licensed at the Faculty of Military
Technology for teaching and research work.
5.1 Topics of Graph Theory
The list of lecture topics on the subject Graph theory,
including hourly subsidies for lectures, exercises and
laboratory exercises in a computer classroom, is given
by the following Table 10.
Table 10.Topics of Graph theory with their hourly
subsidies
#Topic HS L/E/C
1 Basic terminology, basic types of
graphs, simple graphs, degree sequence
4 2/2/0
2 Subgraphs, graph representation,
operations on graphs
4 2/2/0
3 Walks, trails, paths, and cycles in
graphs, connectivity in graphs
4 2/0/0
4 Trees, spanning trees, Cayley’s formula,
Prüfer sequence, Laplacian matrix
4 2/2/0
5 Graph labeling, depth-first search,
breadth-first search
4 2/0/2
6 Isomorphism of graphs and rooted trees,
tree code
4 2/2/0
7 Vertex and edge connectivity, blocks of
a graph, articulation points
4 2/2/0
8 Matchings and covers in bipartite
graphs, perfect matching
4 2/2/0
9 Edge colorings, chromatic index,
Vizing’s theorem
4 2/2/0
10 Vertex colorings, chromatic number,
Brooks’ theorem
4 2/0/2
11 Planar graphs, Kuratowski’s theorem,
Euler’s formula, dual graph
4 2/2/0
12 Eulerian and Hamiltonian graphs,
Chinese postman problem
4 2/2/0
13 Digraphs, basic terminology, digraph
connectivity, Eulerian digraphs
4 2/2/0
14 Flow networks, algorithm for finding a
maximal flow
4 2/2/0
15 Critical Path Method, Project
Evaluation and Review Technique
4 2/0/2
5.2 Classified Credit in Graph Theory
Students are again always introduced to the content of
the classified credit at the beginning of the semester.
At the end of the semester, several sets of solved
preparatory tasks for the exam are made available
to students. The classified credit in Graph theory
consists of a theoretical part and a practical part.
The theoretical part contains four questions about
theory. From 0 to 5 points can be obtained for
each answered question, so a student can receive
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a maximum of 20 points for the theoretical part.
A maximum of another 20 points can be gained from
exercises.
The practical part of the classified credit consists
of the assignment of six tasks. For each task, the
student can get from 0 to 10 points, so for the
practical part, the student can receive a maximum of
60 points. For the classified credit, the student can
get a maximum of 100 points, such as with exams
in the subjects Mathematics I, Mathematics II and
Mathematics III.
5.3 Example of a Classified Credit Test in
Graph Theory
One of the assignments of the classified credit test
had the following theoretical and practical part. The
theoretical part lasts 30 minutes, the practical part
lasts 120 minutes. Thus, the Graph theory classified
credit test took 150 minutes, as was the case with the
Mathematics I, Mathematics II and Mathematics III
exams.
▶.............Thetheoreticalpart .............
A. Define a degree sequence, formulate Havel–
–Hakimi algorithm, and illustrate it in a example.
[5 points]
B. Define an acyclic graph, forest, tree, formulate
statements equivalent to statement that the graph
with nvertices is a tree. [5 points]
C. Define an area of a planar graph, formulate Euler’s
formula and its two consequences for the number
of edges of the planar graph. [5 points]
D. Define a Hamiltonian cycle, a Hamiltonian graph,
a Hamiltonian path, formulate Ore’s theorem.
[5 points]
▶..............Thepracticalpart ..............
1. a) Using the Havel–Hakimi algorithm, show that
the degree sequence D(G) = (4,4,4,3,3,2,2)
is a graph sequence.
b) Draw an example of a graph Gwith a graph
sequence D(G).
c) In the graph G, mark an example of an Eulerian
trail. [10 points]
2. Using the Hungarian algorithm, determine all the
cheapest maximal matchings Mand their cost c
in a bipartite graph G= (V, W ), where V=
{v1, v2, v3, v4}and W={w1, w2, w3, w4}, that
has a cost matrix
C=
4321
1432
1324
4213
.
[10 points]
3. Solve the Chinese postman problem for the
weighted graph Gwith vertices a, b, c, d, e, f, g, h
shown in the figure, whose start and end vertices
are vertex a:
[10 points]
4. a) Determine which two of the graphs F, G, H
are isomorphic and describe the isomorphism
using the bijection of their vertices:
b) Show the plane drawing of the remaining
non-isomorphic graph and verify Euler’s
formula for it.
c) Determine the chromatic number of a
non-isomorphic graph and show an example
of its minimal coloring by marking the vertices
with the numbers. [10 points]
5. In the following network determine the maximum
flow using the Ford–Fulkerson algorithm:
[10 points]
6. Using the CPM method, determine the critical
path, the project completion time Tand the total
time reserves Rcof the individual activities of the
project specified by a network graph with vertices
1,2,3,4,5,6,7and weighted oriented edges:
[10 points]
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5.4 Students’ Success in the Classified Credit
Test in Graph Theory
The following Table 11 presents the success of
students in the classified credit test in Graph theory
in the summer semesters of the academic years
2020/2021, 2021/2022, 2022/2023, cumulatively for
the academic years 2020/2021 to 2022/2023 and
the total number of students who continue to study
successfully.
Table 11.Final evaluations for the classified credit
test in Graph theory in the academic years 2020/2021,
2021/2022, 2022/2023, and from the academic year
2020/2021 to 2022/2023
AY A B C D E FE FFE TN
2020/2021 8 2 3 1 2 0 0 16
2021/2022 12 5 1 1 0 0 0 19
2022/2023 18 7 5 0 0 0 0 30
20/21–22/23 38 14 9 2 2 2 0 65
Figure 4 shows the percentage success rate of
students in the classified credit test in Graph theory
from the academic year 2020/2021 to 2022/2023.
Fig.4: The percentage success rate in the classified
credit test in Graph theory from AY 2020/21 to
2022/23
5.5 Students’ Success in Solving Single Types
of Tasks in the Classified Credit Test in
Graph Theory
The following Table 12 contains the evaluation of
success in the last exam in AY 2022/2023.
Students’ success in the classified credit test in
Graph theory was very high for all the tasks. The
average evaluation of all six types of tasks used in the
academic year 2022/2023 was over 9 points out of 10
possible points.
At the beginning of the semester, students are
again acquainted with the eight types of tasks that are
stated in the second column of Table 12. The third
column contains the number of students who solved
this type of task. The fourth column always shows the
average score, i.e. the number of points that students
achieved from a maximum of 10 points. The average
score received by students on the theoretical part of
the classified credit test on a total of 30 classified
credit tests was 14.13 out of 20 points.
Table 12.Success of students in the task solving
at classified credit test in Graph theory in AY
2022/2023
#Type of exam task NS AS
1 Verification that the degree sequence is
a graph sequence, drawing an example
of such a graph, and determining the
Eulerian trail in this graph.
30 9.67
2 Determination the distance matrix and
the reachability matrix for the graph and
directed graph.
0 —
3 Determination of the number of spanning
trees of the graph using subdeterminants
of the incidence matrix and using
Kirchhoff’s theorems.
0 —
4 Determination all the cheapest maximum
matchings in a bipartite graph with a
given cost matrix using the Hungarian
algorithm.
30 9.13
5 Solving the problem of the Chinese
postman with a given weighted graph
and determining the length of the shortest
walk covering all the edges.
30 9.30
6 Finding isomorphic graphs, plane
drawing of the graph, verifying the
Euler’s formula, determining the
chromatic number of the graph and
its minimum vertex coloring.
30 9.20
7 Determination and drawing the maximum
flow in the network using the Ford–
–Fulkerson algorithm.
30 9.80
8 Determination of the critical path, the
crash duration and the total time reserves
of individual project activities by the
CPM method.
30 9.60
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6 Discussion
In the following subsections, we will present a list
of tasks in the semester exams from Mathematics I,
Mathematics II and Mathematics III and in the
classified credits from Graph theory, where students
achieved the worst and best results, along with the
likely causes.
Let us remark that the originally weak results in
answering the theoretical questions in the theoretical
part of examinations were mostly improved with the
progressing semesters (4.86, 6.85, 5.47 and 14.13
points out of 20 points), as students gradually got used
to studying the theory and not only practical tasks, and
they began to study the theory of greater importance.
In general, unfortunately, the success of students
in semester exams in the subjects Mathematics I,
Mathematics II and Mathematics III has a declining
tendency. This is probably due to weaker knowledge
of high school mathematics for students who had a
lower proportion of direct teaching in high school and
studied some parts of mathematics at home during the
Covid-19 pandemic.
A certain reason is the decreasing number of
Vietnamese students, whose knowledge of secondary
school mathematics is traditionally at a high level
and whose application and diligence exceed those
qualities of an average Czech student.
On the other hand, the success of the classified
credit from the subject Graph theory is high and
relatively permanent, with percentage success rate of
grade A around 60%.
This may be due to the relative popularity of
the subject Graph theory, students’ awareness of
many applications of this subject in the Cybersecurity
specialization, and a more responsible attitude of
students of the 4th semester to study.
6.1 The Least and Most Successful Types of
Tasks in the Mathematics I Exam
Solving a task concerning a linear map has the least
success rate (3.17 points out of 10 points), because a
linear map is the last topic which is not quite easy and
which students do not sufficiently experienced and
practice. The task to verify whether a given relation is
an equivalence relation, also has a very small success
rate (3.67 pts), because it is difficult for students to
prove the reflexivity, symmetry and transitivity of a
given relation based on the property it is defined.
Students achieve the best results (6.95 pts) when
solving a task from propositional logic, which they
usually got to know already in secondary school, and
a task concerning matrix operations (6.23 pts) because
the operations with matrices represent a relatively
simple practical task for students.
6.2 The Least and Most Successful Types of
Tasks in the Mathematics II Exam
The least success rate has the task concerning
polynomial factor rings (2.31 pts), because it is
the last and most difficult topic for students and
the task to determine the orthogonal basis, along
with the task concerning geometrical applications of
different types of vector products (4.33 pts), because
some students have problem with the Gram–Schmidt
process formulas and because some students have a
weak plane and space power of visualization.
The students achieve the highest scores by solving
the task concerning isomorphic algebraic structures
and their properties (7.52 pts), because this topic has
greater attention and sufficient time subsidy. Also,
the task to determine the GCD and LCM of two
integers, along with the task of solving a system of
linear congruence equations, achieve a relatively high
score (6.75 pts), because the students the notions
GCD and LCM, just as the elimination method for
solving linear systems of equations, know from the
secondary school.
6.3 The Least and Most Successful Types of
Tasks in the Mathematics III Exam
The tasks from the first part of the lectures concerning
logic and finite-state machines were very successful.
The highest score (9.88 pts) achieved a relatively
simple task for the finite-state machine and the
decision of which words the finite-state machine
recognizes. High scores (9.67 pts) were achieved
in solving tasks from logic, which the students had
already acquired in the subject Mathematics I, and
minimization of formulas.
On the contrary, the tasks of the second part of
the lectures concerning the function of real variables,
differential and integral calculus and infinite series
managed students with fewer or greater problems.
The lowest score (3.27 pts) was achieved by the
students to solve the tasks on the integral calculus
and its geometric application and the tasks of
differential calculus concerning local extrema and
function properties (3.71 pts).
6.4 The Least and Most Successful Types of
Tasks in the Graph Theory Classified
Credit
Based on a 5-year experience of teaching the subject
Graph theory, the author may state that students
are more popular with this subject than most of
the classical topics from discrete mathematics and
differential and integral calculus. The popularity of
this subject corresponds to the very nice evaluation
that students achieve in the classified credits.
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Students have solved all six types of practical tasks
very well, so it is not possible to talk about tasks with
the smallest or greatest success. The lowest average
score (9.13 pts) was received by the students for the
task of the cheapest maximum matching solved by
the Hungarian algorithm. The highest score (9.80
pts) students achieved in the task determining the
maximum flow using the Ford–Fulkerson algorithm.
7 Conclusion
The primary task and mission of mathematics is to
find solutions to problems that arise in engineering
and applied sciences, such as Cybersecurity. In
conclusion, it can be stated that the study of
the subjects Mathematics I, Mathematics II and
Mathematics III is quite challenging for students
specializing in Cybersecurity. There are several
reasons. One of the most fundamental is the
decreasing level of knowledge of high school
mathematics with which students come to the
Faculty of Military Technology, as well as a less
responsible approach to studies at first. For students,
their studies and preparation for semester exams
are often complicated by bad study habits, little
diligence and willingness to study mathematics and
solve homework examples independently and with
understanding, and little or almost no ability to work
with recommended literature.
Study texts for the subjects Mathematics I, II, III
and Graph theory are continuously processed by the
author of this article and are accessible to students
in electronic form in the Moodle environment.
Currently, the study text Mathematics I has already
been published, [30] and other titles in printed
form will gradually follow. For teaching Graph
theory, we temporarily, with the permission of the
author, use the study text,[31].
As a rule, with the passage of time, students’ study
habits improve and their responsibility in studying
and preparing for exams increases. At the same
time, during the first year, students supplement
their knowledge of high school mathematics, mainly
thanks to mentoring, which takes place every week
in the afternoon for weaker students at the Faculty of
Military Technology. Mentoring, both individual and
group, has been introduced to the faculty for the third
year. It can be stated that this form of supplementary
teaching has proven itself very well and is considered
beneficial even by the students themselves.
Area of Further Development
The author wants to continue monitoring the
success of Cybersecurity students while studying
mathematics, because five years of teaching
experience is not a very long time for any more
fundamental conclusions.
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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2024.21.18
Potůček R.
E-ISSN: 2224-3410
169
Volume 21, 2024
