Discrete Mathematics

Lecture 1Abdul Hafeez

About Me

• PhD – Hamdard University (Research Candidate)

• MS Software Engineering – Hamdard University

• 2nd Year in SMIU (Assistant Professor)• Teaching and developing software From 2003 • Chief Software Consultant at NTS from 2005 • Research Area: SE, Ontology , Formal SE, OOAD

COURSE INFORMATION

Course Title Discrete Mathematical Structures

Course Code MT103

Credit Hours 3(3+0)

Pre-requisite None

Type of Course Core

Academic Year Spring 2013-14

COURSE DESCRIPTION

• Introductory course in discrete mathematics.– Introduce techniques from discrete mathematics

that are widely used in computer science. – How to think logically and mathematically and

apply these techniques in solving problems. – learn logic and proof, sets, functions, algorithms

mathematical reasoning, relations, graphs, trees, and computability are covered in this course.

COURSE GOALS

• Familiarize students with mathematical arguments

• Prove simple arguments• Learn how to develop recursive algorithms

based on mathematical induction• Acquire Knowledge of tree, graph• Learn how to apply discrete mathematics in

problem solving

COURSE LEARNING OUTCOMES

• List and show familiarity with discrete structures such as sets, functions, relations, and sequences.

• Construct mathematical arguments using logical connectives and quantifiers.

• Demonstrate the ability to solve problems using counting techniques and combinatorics

• Justify correctness of an argument using propositional and predicate logic and truth tables.

• Differentiate and distinguished proofs using direct proof, proof by contradiction

• Illustrate how to compute complexity of algorithm

COURSE CONTENTS AND TOPICS ( Week by Week )

Week Topic Covered

1 Logic, Truth Tables, Laws Of Logic

2 Biconditional, Argument, Applications Of Logic

3 Set Theory, Venn Diagram, Set Identities

4 Relations, Types Of Relations Matrix Representation Of RelationsTest/Quiz

COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…)

5 Inverse Of Relations, Functions, Types Of Functions

6 Inverse Function, Composition Of Functions

7 Sequence And Series

8 Mid Term

COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…)

9 Recursion

10 Mathematical Induction

11 Methods Of Proof, Proof By Contradiction

12 Algorithm, CombinatoricsTest/Quiz

COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…)

13 Permutations, Combinations

14 Tree, Inclusion-Exclusion Principle

15 Introduction To Graphs

16 Final Examination

Class Assignments

Submission Date:Week-3 Assignment

Week-6 Assignment

Week-9 Assignment

COURSE TEXTBOOK, RECOMMENDED READINGS

• Textbook– Discrete Mathematics and Its Applications, 6th

edition; by Rosen; McGraw-Hill; 0-07-242434-6.• Reference Book(s) – Discrete Mathematics by Richard Johnsonbaugh,

Prentice Hall, 0135182425.– Discrete Mathematical Structures, 4th Edition, by

Kolman, Busby & Ross, 2000, Prentice-Hall, ISBN: 0-13-0

STUDENT EVALUATION

Course Work Evaluation (Tests / Quizzes /

Assignments / Project)

40

Midterm20

Final40

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 guidlines)– 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

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

Mathematics

• Abstract study of – quantity (numbers)– Structure– Space– Change

Mathematics (Cont…)

• Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written.

• It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.

Mathematics (Cont…)

• Mathematics is used throughout the world as an essential tool in many fields, including – Natural Science ( astronomy, biology, chemistry, Earth

sciences and physics)– Engineering– Medicine– Finance– Social sciences (economics, political Science,

psychology and sociology)

Applied mathematics

Vehicle Routing Problem Example

•Concerned with application of mathematical knowledge to other fields.•Makes use of new mathematical discoveries, such as statistics and game theory

Pure Mathematics

• Without having any application in mind.• There is no clear line separating pure and

applied mathematics• and practical applications for what began as

pure mathematics are often discovered.

Calculus

• A stone, or concretion, formed in the gallbladder, kidneys, or other parts of the body.

• A particular method or system of calculation or reasoning.

• Calculus is the study of how things change.• It provides a framework for modeling systems

in which there is change and a way to deduce the predictions of such models e.g. concept of speed of motion

Calculus (Cont…)

• Calculus is the study of change, with the basic focus being on– Rate of change– Accumulation

Calculus Application• Credit card companies use calculus to set the minimum

payments, considering multiple variables such as changing interest rates and a fluctuating available balance.

• Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed.

• An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart.

Calculus Application (Cont..)

• An architect will use integration to determine the amount of materials necessary to construct a curved dome over a new sports arena

• Statisticians will use calculus to evaluate survey data to help develop business plans for different companies. calculus allows a more accurate prediction for appropriate action.

Calculus Application (Cont..)

• A physicist uses calculus to find the center of mass of a sports utility vehicle to design appropriate safety features that must adhere to federal specifications on different road surfaces and at different speeds.

• An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits.

Discrete Mathematics

• Deals with discrete objects. • Discrete objects are those which are separated

from (not connected to/distinct from) each other.

• Integers (aka whole numbers), automobiles, houses, people etc. are all discrete objects.

• On the other hand real numbers which include irrational as well as rational numbers are not discrete.

Why Discrete Mathematics

• Discrete mathematics is the mathematical language of computer science– Boolean algebra ->Circuits – Logic -> Logic Programming, AI, Software Verification– Set Theory-> DBMS– Tree, Graph -> Algorithms, Networks, Data

Communication – Automata -> Programming Languages– Sequence and Series -> Cryptography, etc