2. Discrete Mathematics Lecture 1 Abdul Hafeez

3. About Me PhD – Hamdard University (Research Candidate) MS Software Engineering – Hamdard University 2 nd 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

4. 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

5. 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.

6. 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

7. 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

8. COURSE CONTENTS AND TOPICS ( Week by Week ) WeekTopic Covered 1Logic, Truth Tables, Laws Of Logic 2Biconditional, Argument, Applications Of Logic 3Set Theory, Venn Diagram, Set Identities 4Relations, Types Of Relations Matrix Representation Of Relations Test/Quiz

9. COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…) 5Inverse Of Relations, Functions, Types Of Functions 6Inverse Function, Composition Of Functions 7Sequence And Series 8Mid Term

10. COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…) 9Recursion 10Mathematical Induction 11Methods Of Proof, Proof By Contradiction 12Algorithm, Combinatorics Test/Quiz

11. COURSE CONTENTS AND TOPICS ( Week by Week ) (Cont…) 13Permutations, Combinations 14Tree, Inclusion-Exclusion Principle 15Introduction To Graphs 16Final Examination

12. Class Assignments Submission Date: Week-3Assignment Week-6Assignment Week-9Assignment

13. 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

14. STUDENT EVALUATION Course Work Evaluation (Tests / Quizzes / Assignments / Project) 40 Midterm 20 Final 40

15. 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

16. 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

17. Mathematics Abstract study of – quantity (numbers) – Structure – Space – Change

18. 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.

19. 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)

20. 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

21. 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.

22. Calculus A stone, or concretion, formed in the gallblad der, 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

23. Calculus (Cont…) Calculus is the study of change, with the basic focus being on – Rate of change – Accumulation

24. 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.

25. 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.

26. 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.

27. 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.

28. 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