ICS-252 Discrete Structure II

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University of Hail

College of Computer Science and Engineering

Dept of Computer Science & Software Engineering

First Semester 2015

ICS-252 Discrete Structure II

ail. S.hagahmoodi@uoh.edu.saOffice P16/8

Dr. Salah Omer Hagahmoodi

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Course Overview

Acknowledgment goes to

Dr. M. Rana

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3 ICS 252 Discrete Structures II (3-0-3)

Number Theory: Modular Arithmetic, Integer Representation, Fermat’s Little

Theorem, Chinese Remainder Theorem, RSA.; Proof Techniques: Methods of

Proofs, Applications from Number Theory, Recursive Definitions; Algorithm

Correctness; Relations: Closures and Equivalence Relations, Partial Orderings

and Lattices, Hasse Diagrams; Recurrence Relations and Generating

Functions; Automata Theory: Finite State Machines, Regular Expressions,

DFA, NDFA and their equivalence, Grammars and Chomsky Hierarchy,

Introduction to Turing Machines.; Abstract Algebra: Groups, Homomorphisms

and Lagrange's Theorem, Applications.

Prerequisite: ICS 102

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Course Objectives

This course introduce the mathematical concepts that form

the basis of computer science, including number theory

(such

as

division,

modular

arithmetic,

integer

representation,

Fermat’s

Little

Theorem,

Chinese

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1-5

A student completing this module unit should be

able to:

1- Understand what discrete structures are, and

why it is important in computer science.

2- Understand the basic concept related to

i.

number theory such as division, congruent,

prime numbers, gcd, and relatively primes.

ii. different b-systems to represent the integers.

iii. different techniques of proof.

iv. different types of relations.

v. what is the mean by recursive?

vi. basic concepts in automata theory.

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3- Ability to:

i. use division and modular arithmetic for generating

pseudorandom numbers, hash function and cryptology.

ii. prove some theorems related to number theory.

iii.change a decimal integer to other b-system and vice

via.

iv.prove a mathematical statement using of the method

proof.

v. prove that the algorithm is correct.

vi.solve many exercise related to studied topics.

vii.

use Fermat’s little and Chinese Remainder theorem in

RSA.

viii.determine the language generated by deterministic

finite automata and regular expression

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Course Assessment

Assessment: Assessment for this course will be based on homework, class tests (or quizzes), major exams, a final exam as in the following:

Major 1 20%

Major 2 20%

Quizzes + HW 15%

Final Exam 45%

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Lecture one: Outlines

• Define Discrete Structures

• Importance of Discrete Structures

• What is Number Theory

• Number Theory Areas

• Integer Representation

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Discrete Structures

Definition: Discrete structure deals with discrete objects.

Discrete objects are those which are separated from (not connected) each other.

Examples:

 Integers (whole numbers 5, 10, 15), rational numbers (ones that

can be expressed as the quotient of two integers i.e. 10/5) are discrete object.

 Automobiles, houses, people etc. are all discrete objects.

 On the other hand real numbers (such as 5.35, 5.2345) which

include irrational as well as rational numbers are not discrete.

 Between any two different real numbers there is another real

number different from either of them. So they are packed without any gaps and can not be separated from their immediate

neighbors. In that sense they are not discrete.

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Importance of Discrete Structures

• It provides foundation material for computer science.

• It includes important material from such areas as set theory, logic, graph theory, and combinatorics.

• It enables students to create and understand a proof— either a formal symbolic proof or a less formal but still

mathematically rigorous argument—is essential in formal

specification, in verification, in databases, and in

cryptography.

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Importance of Discrete Structures

• The graph theory concepts are used in networks, operating systems, and compilers.

• Set theory concepts are used in software engineering and in databases.

• In engineering, It can be used to control multiproduct batch plants, synthesis of new multifunctional reactors

and design of a new class of simulator.

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What is Number Theory?

Definition: Number theory

is the branch of pure

mathematics concerned with the properties of

numbers in general, and integers in particular, as

well as the wider classes of problems that arise

from their study.

Number Theory Areas:

• Elementary number theory-

Study of Integers.

• Analytic number theory-

employs the machinery

of calculus and complex analysis to tackle

questions about integers.

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Number Theory Areas

13 • Algebraic number theory- the concept of a number is

expanded to the algebraic numbers which are roots of polynomials with rational coefficients.

• Geometry of numbers- incorporates some basic

geometric concepts, such as lattices, into number-theoretic questions.

• Combinatorial number theory- deals with number

theoretic problems which involve combinatorial ideas in their formulations or solutions.

• Computational number theory- studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important

applications in cryptography.

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What is an Integer

• The integers from the Latin integer, literally are

"untouched", hence we can say "whole“.

• formed by the natural numbers including 0 (0, 1, 2, 3, ...)

together with the negatives of the non-zero natural numbers (−1, −2, −3, ...).

• They are numbers that can be written without a fractional or decimal component, and fall within the set {... -3 ,−2, −1, 0, 1, 2, 3...}.

• For example, 65, 7, and −756 are integers; 1.6 and 1½ are

not integers. The set of all integers is often denoted by a boldface Z (or blackboard bold ℤ), which stands for Zahlen

(German for numbers, pronounced [ˈtsaːlən])

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Integer Representation

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• Integers can be thought of as discrete, equally spaced points on an infinitely long number line.

• In everyday life we use decimal notation (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to express integers. For Example: 965 is used to denote as;

• In computers we normally use Binary (0, 1) number system (with 2 as the base) e.g. 10111.

• Octal (base 8)- (0, 1, 2, 3, 4, 5, 6, 7) number system, e.g 457

• Hexadecimal (base 16)- (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) e.g. 05DF

By using these number systems we can represents characters, such as letters upper and lower cases and numbers.

ICS-252 Dr. Salah Omer Hagahmoodi, Assistant Professor, CSSE, University of Hail.