Lecture1_Introduction.pptx by doctor ahikisKye Emmanuel

Introduction to discrete maths

1. Introduction TO DISCRETE MATHEMATICS 1 *Acknowledgment: This material is derived and adapted from “Discrete Mathematics and its Applications, Seventh Edition”, Kenneth H. Rosen

2. About the Instructor Name: Dr. Emmanuel Ahishakiye (PhD) Office Faculty of science, Next after Department of computer science office (to the left). Contacts Tel: [phone] Email: [email protected] 2

3. Course Details Course Code and Name: SCS 1104 Discrete Mathematics/Discrete Mathematics Structures. Course Level: Year 1 Semester 1 Course Credit: 3/ 4CU Contact Hours: 45/ 60 3

4. Course description The purpose of this course is to understand and use discrete structures that are backbones of computer science. In particular, this class is meant to introduce logic, proofs, sets, relations, functions and counting, with an emphasis on applications in computer science. 4

5. Detailed Course Outline (1) Number Systems: Decimal, Binary, Octal, hexadecimal, their conversions and applications in digital machines. Logic: propositional logic, logical equivalence, predicates & quantifiers, and logical reasoning. Boolean Algebra: Boolean variables, addition, multiplication, functions, laws, equivalences and applications. 5

6. Detailed Course Outline (2) Discrete Structures: set, functions and their applications. Counting: permutations, combinations and applications. Recursion: recurrence relations, solving recurrence relations and their applications in computer science. Tree Data Structures: notations used in trees, types of trees, tree traversals, and applications in data organization. 6

7. Main Course Reference Kenneth Rosen. Discrete Mathematics and Its Applications, 7th Edition, McGraw Hill Publishing Co., 2012. “Discrete Mathematical Structures” by B. Kalman Prentice Hall (1996). “Discrete Mathematics with Application” by Susana. 7

8. Assignments There will be several assignments throughout up to the completion of the course. The assignments are due at the time specified by the instructor. In general, no extensions will be granted. Incase of issues or challenges, communicate as early as possible. 8

9. Academic Honesty All the work in this course should be done either independently or in groups. Collaborations on homeworks are not permitted. Cheating and any other anti intellectual behavior, including giving your work to someone else, will be dealt with severely. Plagiarism will not be tolerated. Whenever you use other peoples work (on internet, text books etc), you need to acknowledge the source accordingly. If you feel you may have violated the rules speak to the instructor as soon as possible. 9

10. Mode of Delivery & Assessment Mode of Delivery This course is taught by lectures, assignments and students presentations. Assessment Assignments : 30% Attendance : 5% Participation in class : 5% Final course Examination : 60% Total : 100% 10

11. Introduction Discrete mathematics describes processes that consist of a sequence of individual steps. This contrasts with calculus, which describes processes that change in a continuous fashion. Whereas the ideas of calculus were fundamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age. Numerical Data can be  Discrete  or  Continuous : Discrete data  is counted, Example: the number of students in a class. Continuous data  is measured. Example: A person's height: could be any value (within the range of human heights), not just certain fixed heights. 11

12. Discrete and Continuous 12

13. Introduction Dfn 1: Discrete mathematics  is the study of  mathematical structures  that are fundamentally  discrete  rather than continuous. Dfn 2: Discrete Mathematics is a collection of mathematical topics that examine and use finite or countably infinite mathematical objects. 13

14. Problems solved by Discrete Mathematics (1) How many ways are there to choose a valid password on a computer system? What is the probability of winning a lottery? Is there a link between two computers in a network? How can I identify spam e-mail messages? How can I encrypt a message so that no unintended recipient can read it? What is the shortest path between two cities using a transportation system? 14

15. Problems solved by Discrete Mathematics (2) How can a list of integers be sorted so that the integers are in increasing order? How many steps are required to do such a sorting? How can it be proved that a sorting algorithm correctly sorts a list? How can a circuit that adds two integers be designed? How many valid Internet addresses are there? 15

16. Goals of a Discrete Mathematics (1) Mathematical Reasoning: Students must understand mathematical reasoning in order to read, comprehend, and construct mathematical arguments. The technique of mathematical induction is stressed through many different types of examples of such proofs and a careful explanation of why mathematical induction is a valid proof technique. Combinatorial Analysis: An important problem-solving skill is the ability to count or enumerate objects. To understand this concept well, the basic techniques of counting should be covered. The stress is on performing combinatorial analysis to solve counting problems and analyze algorithms, not on applying formulae. 16

17. Goals of a Discrete Mathematics (2) Discrete Structures: A course in discrete mathematics should teach students how to work with discrete structures, which are the abstract mathematical structures used to represent discrete objects and relationships between these objects. These discrete structures include sets, permutations, relations, graphs, trees, and finite-state machines. Algorithmic Thinking : Certain classes of problems are solved by the specification of an algorithm. After an algorithm has been described, a computer program can be constructed implementing it. The mathematical portions of this activity, which include the specification of the algorithm, the verification that it works properly, and the analysis of the computer memory and time required to perform it, are all covered in discrete mathematics. 17

18. Goals of a Discrete Mathematics (3) Applications and Modeling : Discrete mathematics has applications to almost every conceivable area of study. There are many applications to computer science and data networking in this text, as well as applications to such diverse areas as chemistry, biology, linguistics, geography, business, and the Internet. These applications are natural and important uses of discrete mathematics and are not contrived. Modeling with discrete mathematics is an extremely important problem-solving skill, which students have the opportunity to develop by constructing their own models. 18

19. Applications of Discrete Maths in Computer Science Discrete Mathematics covers some important concepts such as set theory, graph theory, logic, permutation and combination as well which are essential in computing. It is therefore used in programming languages, software development, cryptography, algorithms etc. 19

20. Questions? 20