1. The document outlines discrete mathematics competencies covered at different levels in the undergraduate curriculum at Saint-Petersburg Electrotechnical University.
2. Many competencies are covered in the discrete mathematics course in the first year, while others are covered in courses like mathematical logic and algorithm theory in later years.
3. LETI aims to develop additional competencies beyond the SEFI levels, such as skills in mathematical logic, graphs, algorithms, and finite state machines.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
Now we have learnt the basics in logic.
We are going to apply the logical rules in proving mathematical theorems.
1-Direct proof
2-Contrapositive
3-Proof by contradiction
4-Proof by cases
A New Approach of Learning Hierarchy Construction Based on Fuzzy LogicIJERA Editor
In recent years, adaptive learning systems rely increasingly on learning hierarchy to customize the educational logic developed in their courses. Most approaches do not consider that the relationships of prerequisites between the skills are fuzzy relationships. In this article, we describe a new approach of a practical application of fuzzy logic techniques to the construction of learning hierarchies. For this, we use a learning hierarchy predefined by one or more experts of a specific field. However, the relationships of prerequisites between the skills in the learning hierarchy are not definitive and they are fuzzy relationships. Indeed, we measure relevance degree of all relationships existing in this learning hierarchy and we try to answer to the following question: Is the relationships of prerequisites predefined in initial learning hierarchy are correctly established or not?
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Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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