Discrete Mathematics

Saint-Petersburg Electrotechnical

University «LETI»

Discrete Math in LETI undergraduate curricula

Year 1 Discrete Math

Year 2Math Logic and

Algorithm Theory

Year 3

Year 4

SEFI levels

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

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

SEFI Level 1Content 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

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

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

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)

SEFI Level 1Content Competence

Graphsrecognise a graph (directed and/or undirected) in a real Situation

understand the notions of a path and a cycle

In addition at LETIContent Competence

Graphs Obtain an incidence matrix for a graph

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...

In LETI (although those competences are applied for Statistics and Probability too) you obtain them studying

Discrete Mathematics.

In addition at LETIContent 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

SEFI Level 2

Content Competence

Number systemscarry out arithmetic operations in the binary system

carry out arithmetic operations in the hexadecimal system

In addition at LETI

Content Competence

Number systemscarry out arithmetic operations in any number system

convert numbers from any number system to deimal and vice versa

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

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

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

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 LETIContent Competence

RelationsObtain a the graph and his matrix for a relation

Use topological sort algorithm and transitive closure algorithms

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

GraphsFind the distance (shortest way) between two vertices in a graph

Recognize planar graph

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

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

SEFI Level 3Some 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

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

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