This document provides an introduction to the Discrete Structures course MAT-104. It includes the course objectives, which are to introduce topics like logic, set theory, functions, and graph theory. Active learning strategies like assignments and projects will be used. The course will be evaluated through quizzes, assignments, presentations, and a midterm and final exam. Policies around attendance and plagiarism are also outlined.
This slides contains assymptotic notations, recurrence relation like subtitution method, iteration method, master method and recursion tree method and sorting algorithms like merge sort, quick sort, heap sort, counting sort, radix sort and bucket sort.
This slides contains assymptotic notations, recurrence relation like subtitution method, iteration method, master method and recursion tree method and sorting algorithms like merge sort, quick sort, heap sort, counting sort, radix sort and bucket sort.
This presentation contains:
1. Introduction
2. Central areas of TOC
3. Complexity theory
4. Computability theory
5. Automata theory
6. Related terminologies
7. Strings
8. Languages
9. Proof, Theorem, Lemma, Corollaries
Design & Analysis of Algorithms Lecture NotesFellowBuddy.com
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Hey friends, here is my "query tree" assignment. :-) I have searched a lot to get this master piece :p and I can guarantee you that this one gonna help you In Sha ALLAH more than any else document on the subject. Have a good day :-)
We will discuss the following: Graph, Directed vs Undirected Graph, Acyclic vs Cyclic Graph, Backedge, Search vs Traversal, Breadth First Traversal, Depth First Traversal, Detect Cycle in a Directed Graph.
This presentation contains:
1. Introduction
2. Central areas of TOC
3. Complexity theory
4. Computability theory
5. Automata theory
6. Related terminologies
7. Strings
8. Languages
9. Proof, Theorem, Lemma, Corollaries
Design & Analysis of Algorithms Lecture NotesFellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Hey friends, here is my "query tree" assignment. :-) I have searched a lot to get this master piece :p and I can guarantee you that this one gonna help you In Sha ALLAH more than any else document on the subject. Have a good day :-)
We will discuss the following: Graph, Directed vs Undirected Graph, Acyclic vs Cyclic Graph, Backedge, Search vs Traversal, Breadth First Traversal, Depth First Traversal, Detect Cycle in a Directed Graph.
In a world of data explosion, the rate of data generation and consumption is on the increasing side, there comes the buzzword - Big Data.
Big Data is the concept of fast-moving, large-volume data in varying dimensions (sources) and
highly unpredicted sources.
The 4Vs of Big Data
● Volume - Scale of Data
● Velocity - Analysis of Streaming Data
● Variety - Different forms of Data
● Veracity - Uncertainty of Data
With increasing data availability, the new trend in the industry demands not just data collection,
but making ample sense of acquired data - thereby, the concept of Data Analytics.
Taking it a step further to further make a futuristic prediction and realistic inferences - the concept
of Machine Learning.
A blend of both gives a robust analysis of data for the past, now and the future.
There is a thin line between data analytics and Machine learning which becomes very obvious
when you dig deep.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
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How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
3. COURSE OBJECTIVES
Mathematical reasoning: introduction to logic, propositional and
predicate calculus; negation disjunction and conjunction; implication
and equivalence; truth tables; predicates; quantifiers; natural
deduction; rules of Inference; methods of proofs; use in program
proving; resolution principle; Set theory: Paradoxes in set theory;
inductive definition of sets and proof by induction; Relations,
representation of relations by graphs; properties of relations,
equivalence relations and partitions; Partial orderings; Linear and
wellordered sets; Functions: mappings, injection and surjection,
composition of functions; inverse functions; special functions; Peano
postulates; Recursive function theory; Elementary combinatorics;
counting techniques; recurrence relation; generating functions. Graph
Theory: elements of graph theory, Planar Graphs, Graph Colouring,
Euler graph, Hamiltonian path, trees and their applications.
4. LEARNING STRATEGIES
Active learning strategies are employed in this course to encourage students'
participation in class and to foster their abilities to gather information and data
from its sources and analyze it.
Active learning strategies include assignments/projects where students work in
individual and in teams to solve certain problems. Readings may include many
Research Papers and other related studies.
5. RECOMMENDED RESOURCES
• Discrete Mathematical Structure with Application to
Computer Science, J. P. Temblay and B Manohar, McGraw-
Hill, 2nd Edition.
• Discrete Mathematics, 7th edition, Richard Johnson Baugh,
2008, Prentice Hall Publishers.
• Relevant Research Papers (Journal + Conference)
8. ATTENDANCE POLICY
• Students are expected to attend their classes. Absence never exempts a
student from the work required for satisfactory completion of the courses.
• Excessive absences of any course will result in:
First warning for absence of 10% of the class hours
Second warning for absence of 20% of the class hours
• A failing grade in the course for an absence of 25% of the class hours (as per
HEC guidelines)
• Exception to (3) may be made in the case of serious illness or death to an
immediate family member if approved by the dean of the college. In such
case, the student will receive a W grade in the course
9. PLAGIARISM
It is use of someone else’s idea, words, projects, artwork, phrasing, sentence
structure, or other work without properly acknowledging the ownership
(source) of the property.
Plagiarism is dishonest because it misrepresents the work of someone else as
ones own.
Students who are suspected of plagiarism will answer to an investigation
Those found guilty will face a disciplinary action as per the university
rules.
10. ASSESSMENT BREAKDOWN(THEORY)
100 MARKS
Quizzes 10%
Two + one quizzes
Assignments 10%
Two+One assignment
Presentation 15%
Class Participation/Attend. 05%
Total 40%
Mid term 20%
Final 40%
13. INTRODUCTION
Discrete Structure describes processes that consist
of a sequence of individual steps, as compared to
forms of mathematics that describe processes that
change in a continuous manner. The major topics we
cover in this course are single-membership sets,
mathematical logic, induction, and proofs.
14. WHATISDISCRETEMATHEMATICS?
• Discrete mathematics focuses on problems that are not
over a continuous domain. For example, is it possible to
visit 3 islands in a river with 6 bridges without crossing
any bridge more than once? That is a discrete math
problem (because there are a finite (fixed, discrete)
number of bridges). Or, what is the smallest number of
telephone lines needed to connect 200 cities? The
numbers can be large and the logic can be complex, but
these type of problems are different from finding an
optimal value for a function where the domain can be 3,
16. WHY DISCRETE STRUCTURE /
MATHEMATICS? (I)
Computers use discrete structures to represent
and manipulate data.
Computer Science is not Programming
Computer Science is not Software Engineering
Edsger Dijkstra: “Computer Science is no more
about computers than Astronomy is about
telescopes.”
Computer Science is about problem solving.
17. WHY DISCRETE MATHEMATICS?
(II)
Mathematics is at the heart of problem solving
Defining a problem requires mathematical rigor
Use and analysis of models, data structures,
algorithms requires a solid foundation of
mathematics
To justify why a particular way of solving a
problem is correct or efficient (i.e., better than
another way) requires analysis with a well-defined
mathematical model.
18. PROBLEM SOLVING REQUIRES MATHEMATICAL
Your boss is not going to ask you to solve
an MST (Minimal SpanningTree) or
aTSP (Travelling Salesperson Problem)
Rarely will you encounter a problem in an abstract
setting
However, he/she may ask you to build a rotation
of the company’s delivery trucks to minimize fuel
usage
It is up to you to determine
a proper model for representing the problem and
a correct or efficient algorithm for solving it
19. Why Discrete Math?
Design efficient computer systems.
•How did Google manage to build a fast search engine?
•What is the foundation of internet security?
algorithms, data structures, database,
parallel computing, distributed systems,
cryptography, computer networks…
Logic, sets/functions, counting, graph theory…
20. What is discrete mathematics?
logic, sets, functions, relations, etc
Logic: artificial intelligence (AI), database, circuit design
Counting: probability, analysis of algorithm
Graph theory: computer network, data structures
Number theory: cryptography, coding theory
21. L
OGIC
Logicis the study of the principles and methods
that distinguishes between a valid and an invalid
argument.
23. Topic 1: Logic and Proofs
Logic: propositional logic, first order logic
Proof: induction, contradiction
How do computers think?
Artificial intelligence, database, circuit, algorithms
26. Topic 3: Graph Theory
• Relations, graphs
• Degree sequence, isomorphism, Eulerian graphs
• Trees
Computer networks, circuit design, data structures
27. Topic 4: Number Theory
• Number sequence
• Euclidean algorithm
• Prime number
• Modular arithmetic
Cryptography, coding theory, data structures
28. PROPOSITIONS
A statement that has a truth value
Which of the following are propositions?
TheWashington State flag is red
It snowed inWhistler, BC on January 4, 2008.
Hillary Clinton won the democratic caucus in Iowa
Space aliens landed in Roswell, New Mexico
Ron Paul would be a great president
Turn your homework in onWednesday
Why are we taking this class?
If n is an integer greater than two, then the equation an + bn = cn has no
solutions in non-zero integers a, b, and c.
Every even integer greater than two can be written as the sum of two
primes
This statement is false
– Propositional variables: p, q, r, s, . . .
– Truth values: T for true, F for false