1. Introduction to Discrete Mathematics AB C a = qb+r gcd(a,b) = gcd(b,r)

2. 2 210241 Discrete Mathematics

3. Prerequisites:Basic Mathematics Companion Course :--- Course Objectives: To introduce several Discrete Mathematical Structures found to be serving as tools even today in the development of theoretical computer science. 1.To introduce students to understand, explain, and apply the foundational mathematical concepts at the core of computer science. 2.To understand use of set, function and relation models to understand practical examples, and interpret the associated operations and terminologies in context. 3.To acquire knowledge of logic and proof techniques to expand mathematical maturity. 4.To learn the fundamental counting principle, permutations, and combinations. 5.To study how to model problem using graph and tree. 6.To learn how abstract algebra is used in coding theory.

4. 4 Textbook Discrete Mathematics and Its Applications by Kenneth H. Rosen

5. Course Outcomes 5 Course Outcomes: On completion of the course, learner will be able to– CO1: Formulate problems precisely, solve the problems, apply formal proof techniques, and explain the reasoning clearly. CO2: Apply appropriate mathematical concepts and skills to solve problems in both familiar and unfamiliar situations including those in real-life contexts. CO3: Design and analyze real world engineering problems by applying set theory, propositional logic and to construct proofs using mathematical induction. CO4: Specify, manipulate and apply equivalence relations; construct and use functions and apply these concepts to solve new problems. CO5: Calculate numbers of possible outcomes using permutations and combinations; to model and analyze computational processes using combinatorics. CO6: Model and solve computing problem using tree and graph and solve problems using appropriate algorithms. CO7: Analyze the properties of binary operations, apply abstract algebra in coding theory and evaluate the algebraic structures.

6. Unit I Contents

7. What is Discrete Mathematics  Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values.  The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatory. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation  The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.  Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits.

8. DISCREATE VS CONTINUES Examples of discrete Data -Number of boys in the class. -Number of candies in a packet. -Number of suitcases lost by an airline. Examples of cont inuous Data -Height of a person. -Time in a race. -Distance traveled by a car.

9. Importance of DM

10. 10  It is a very good tool for improving reasoning and problem-solving capabilities.  It provides ability to comprehend mathematical arguments  Discrete mathematics provides excellent modeling tools for analyzing real-world phenomena that varies in one state or another and is a vital tool used in a wide range of applications, from computers to telephone call routing and from personnel assignments to genetics.  It equips computer science students with logical and mathematical skills. Discrete mathematics is the study of mathematics that underpins computer science, with a focus on discrete structures, for example, graphs, trees and networks.

12. Example 1 How to play Rubik Cube? Google: Rubik cube in 26 steps

13. Why Mathematics? 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, number theory, counting, graph theory…

14. Logic and Proofs Logic: propositional logic, first order logic Proof: induction, contradiction How do computers think? Artificial intelligence, database, circuit, algorithms

15. Number Theory Number sequence (Extended) Euclidean algorithm Prime number, modular arithmetic, Chinese remainder theorem Cryptography, RSA protocol Cryptography, coding theory, data structures

16. Counting Sets and Functions Combinations, Permutations, Binomial theorem Counting by mapping, pigeonhole principle Recursions Probability, algorithms, data structures AB C

17. Counting How many steps are needed to sort n numbers? Algorithm 1 (Bubble Sort): Every iteration moves the i-th smallest number to the i-th position Algorithm 2 (Merge Sort): Which algorithm runs faster?

18. Graph Theory Graphs, Relations Degree sequence, Eulerian graphs, isomorphism Trees Matching Coloring Computer networks, circuit design, data structures

19. Graph Theory How to color a map? How to send data efficiently?

20. Familiar? Obvious? c b a Pythagorean theorem

21. c b a (i) a c  c square, and then (ii) an a  a & a b  b square Good Proof b-a We will show that these five pieces can be rearranged into: b-a And then we can conclude that

22. c c c a b c b-ab-a Good Proof The five pieces can be rearranged into: (i) a c  c square

23. c b a b-a Good Proof How to rearrange them into an axa square and a bxb square?

24. b a a a b-ab-a 74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml b Good Proof

25. Bad Proof A similar rearrangement technique shows that 65=64… What’s wrong with the proof?

26. Mathematical Proof To prove mathematical theorems, we need a more rigorous system. http://en.wikipedia.org/wiki/Pythagorean_theorem Euclid’s proof of Pythagorean’s theorem The standard procedure for proving mathematical theorems is invented by Euclid in 300BC. First he started with five axioms (the truth of these statements are taken for granted). Then he uses logic to deduce the truth of other statements. 1.It is possible to draw a straight line from any point to any other point.straight line 2.It is possible to produce a finite straight line continuously in a straight line.finite 3.It is possible to describe a circle with any center and any radius.circle 4.It is true that all right angles are equal to one another.right angles 5.("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles,Parallel postulateinterior angles the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.intersectangles

27. Know about great philosopher's : Georg Contor, Richard Dedekind, Aristotle More about good topic and nice presentation, than mathematical difficulty. A Project  Interesting or curious problems, interesting history  Surprising or elegant solutions  Nice presentation, easy to understand

28.  Magic tricks  More games, more paper folding, etc  Logic paradoxes  Prime numbers  Game theory Project Ideas