Discrete Mathematics

1. Discrete Mathematics
Saint-Petersburg Electrotechnical
University «LETI»

2. Discrete Math in LETI
undergraduate curricula
Year 1 Discrete Math
Year 2
Math Logic and
Algorithm Theory
Year 3
Year 4

3. SEFI levels

4. Let's use this colouring to distinguish competences which are
treated different ways in LETI
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of semester 1

5. SEFI Core 0
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of Semester 1
Content Competence
Sets
understand the concepts of a set, a subset and the empty set
determine whether an item belongs to a given set or not
use and interpret Venn diagrams
find the union and intersection of two given sets
apply the laws of set algebra

6. SEFI Level 1
Content Competence
Mathematical logic
recognise a proposition
negate a proposition
form a compound proposition using the connectives AND,
OR, IMPLICATION
construct a truth table for a compound proposition
construct a truth table for an implication
verify the equivalence of two propositions using a truth table
identify a contradiction and a tautology
construct the converse of a proposition
obtain the contrapositive form of an implication
understand the unversal quantifier 'for all'
understand the existential quantifier 'there exists'
negate propositions with quantifiers
follow simple examples of direct and indirect proof
follow a simple example of a proof by contradiction

7. In addition at LETI
All those competences should be obtained by LETI students in
the Semester 3, which gives students the ability to work at the
area of ATP (Automatical Theorem proving)
Content Competence
Mathematical logic
recognize prenex and Scolem form of first-order
formulas
obtain prenex and scolem form for a certain formula
unify first-order logic formulas
use resolution method for propositions and first order
logic

8. SEFI Level 1
Content Competence
Sets
understand the notion of an ordered pair
find the Cartesian product of two sets
define a characteristic function of a subset of a given universe
compare the algebra of swithcing circuits to that of set algebra and
logical connectives
analyse simple logic circuits comprising AND, OR, NAND, NOR and
EXCLUSIVE OR gates
understand the concept of a countable set
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of Semester 1

9. SEFI Level 1
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of Semester 1
Content Competence
Mathematical
induction and
recursion
understand (weak) mathematical induction
follow a simple proof which uses mathematical induction
define a set by induction
use structural induction to prove some simple properties of
a set which is given by induction
understand the concept of recursion
define the factorial of a positive integer by recursion (any
other suitable example will serve just as well)

10. SEFI Level 1
Content Competence
Graphs
recognise a graph (directed and/or undirected) in a
real Situation
understand the notions of a path and a cycle
In addition at LETI
Content Competence
Graphs Obtain an incidence matrix for a graph

11. Content Competence
Combinatorics
evaluate the number of ways of arranging unline objects in a line
evaluate the number of ways of arranging objects in a line,
where some are alike
evaluate the number of ways of arranging unlike objects in a ring
evaluate the number of ways of permuting r objects from n
unlike objects
evaluate the number of combinations of r objects from n unlike
objects
use the multiplication principle for combinations
SEFI Level 1
(Statistics and Probability)
In SEFI you obtain all those competences while studying
Staticstics and Probability...

12. In LETI (although those competences are applied for
Statistics and Probability too) you obtain them studying
Discrete Mathematics.
In addition at LETI
Content Competence
Combinatorics
Obtain a generating function for a simple sequence
Find sequence element using generating function
Use inclusion-exclusion principle to obtain a number of elements
in the set
Enumerate permuttations, subsets, k-element subsets of the set

13. SEFI Level 2
Content Competence
Number systems
carry out arithmetic operations in the binary system
carry out arithmetic operations in the hexadecimal system
In addition at LETI
Content Competence
Number systems
carry out arithmetic operations in any number system
convert numbers from any number system to deimal and
vice versa

14. Content Competence
Algebraic operations
use Euclid's algorithm for finding the greates common divisor
understand the notion of a group
establish the congruence of two numbers modulo n
understand and carry out arithmetic operations in Z_n,
especially in Z_2
carry out arithmetic operations on matrices over Z_2
understand the Hamming codes as an application of the
above (any other suitable code will serve just as well).
SEFI Level 2
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of Semester 1

15. In addition at LETI
Content Competence
Algebraic operation
Use Chinese remainder theorem.
Use Euler's phi function and Fermat–Euler theorem or
Euler's totient theorem.
Using of RSA algorithm
Solve linear Diophantine equations with 2 variables

16. Content Competence
Recursion and difference
equations
define a sequence by a recursive formula
obtain the general solution of a linear first-order
difference equation with constant coefficients
obtain the particular solution of a liner first-order
differnce equation with constant coefficients which
satisfies suitable given conditions
obtain the general solution of a linear second-order
difference equation with constant coefficients
obtain the particular solution of a liner second-order
differnce equation with constant coefficients which
satisfies suitable given conditions
SEFI Level 2
For competences, which are included into Discrete Math curriculum
For competences, which are not included in LETI Math curricula
For competences, which are driven out to other Math cources
For competences, which should be obtained by students to the end of Semester 1

17. Content Competence
Relations
understand the notion of binary relation
find the composition of two binary relations
find the inverse of a binary relation
understand the notion of a ternary relation
understand the notion of an equivalence relation on a set
verify whether a given relation is an equivalence relation or not
understand the notion of a partition on a set
view an equivalence either as a relation or a partition
understand the notion of a partial order on a set
understand the differnce between maximal and greatest element,
and between minimal and smallest element
SEFI Level 2
In addition at LETI
Content Competence
Relations
Obtain a the graph and his matrix for a relation
Use topological sort algorithm and transitive closure algorithms

18. Content Competence
Graphs
recognise an Euler trail in a graph and/or an Euler graph
recognise a Hamilton cycle (path) in a graph
find components of connectivity in a graph
find components of strong connectivity in a directed graph
find a minimal spanning tree of a given connected graph
SEFI Level 2
In addition at LETI
Content Competence
Graphs
Find the distance (shortest way) between two vertices in a graph
Recognize planar graph

19. Content Competence
Algorithms
understand when an algorithm solves a problem
understand the 'big O' notantion for functions
understand the worst case analysis of an algorithm
understand one of the sorting algorithms
understand the idea of depth-first search
understand the idea of breadth-first search
understand a multi-stage algorithm (for example, finding the
shortest path, finding the minimal spanning tree or finding
maximal flow)
understand the notion of a polynomial-time-solvable problem
understand the notion of an NP problem (as a problem for
which It is 'easy' to verify an affirmative answer)
understand the notion of an NP-complete problem (as a
hardest problem among NP problems).
SEFI Level 2

20. In addition at LETI
Content Competence
Algorithms understand the notion of Turing machine
run simple turing machines on paper
construct simple Turing machine
run Markov algorithm

21. SEFI Level 3
Some of SEFI Level 3 competences could be obtained
while studying our general courses
Content Competence
Lattices and
Boolean algebra
Understand the concept of Boolean fucntion
Construct a truth table for a function
Obtain CNF and DNF of a function
Obtain Zhegalkin polynom of a function
Build a composition of two or more functions in different forms
Recognize function membership in one of tge Post Classes
Use Post criteria for a set of functions
Grammars and
languages
Recognize context-free grammar
Construct context-free grammar for a simple language
Build a parser for a grammar using Virt algorithm

22. We cannot find those concept and competitions in
SEFI table, even at Level 3. Probably, they are
considered as not subject of Math, but a one of
Computer Science.
In addition at LETI
Content Competence
Finite State Machines
recognize table and graph representation of FSM
recognize automata language
carry out set operations with automata languages
obtain FSM for regular expression and vice versa
obtain determined FSM for non-determined one
FSM minimization

23. Content Competence
Algebra of polynoms
carry out arithmetic operations with polynomials in Z[x], R[x]
and alos modulo n
factorize simple polynomials (including factirzation module n)
find GCD using Euclid algorithm for polynomials
find rational root of a integer polynomial
using Lagrange interpolation formula
In addition at LETI
This competences are necessary to students who will later work
with polynomial codes