introduction fundamentals sets and sequences

Discrete MathematicsCSC 1700-4

Contact Information

Sherzod Turaev

Assistant Professor, Dr.

Department of Computer Science

Kulliyyah of Information & Communication Technology

Office: C3‐21

Email: [email protected]

Web: www.sherzod.info2© S. Turaev, CSC 1700 Discrete Mathematics

ClassesLectures

Time: 11.30 AM – 12.50 PM

Date: Tuesday & Thursday

Location: Level 4C, LR19

Tutorial Classes

Time: 17.00 – 18.50 PM

Date: Thursday

Location: Level 1C, LR1

3© S. Turaev, CSC 1700 Discrete Mathematics

Required ReferenceKolman, Busby, RossDiscrete Mathematical Structures6/E.NJ: Pearson Prentice Hall2013 (2009)


Recommended References1. Rosen, K. (2013) Discrete Mathematics and Its

Applications. 7/E. NY: McGraw Hill.

2. Epp, S. (2011) Discrete Mathematics with Applications. 4/E. Brooks/Cole Cengage Learning.

3. Johnsonbaugh, R. (2009) Discrete Mathematics. 6/E. NJ: Pearson Prentice Hall.


i‐Taleem System

http://italeem.iium.edu.my/

• Lecture Slides/Notes

• Home assignments

• Assessment Results

• Announcements, Discussions, Q&A, etc.


Course Assessments & Marking


METHOD MARKING (%)

Home assignments (5) 10

Quizzes (3) 30

Mid‐term examination 20

Final examination 40

Course OutlineWeek Topics1 Fundamentals

Sets and subsets. Operations on sets. Sequence. Properties of Integers. Matrices.

2‐3 Logic

Propositions and Logical operations. Conditional statements. Methods of proof. Mathematical induction.


Course OutlineWeek Topics4 Counting

Permutations. Combinations. Pigeonhole principle. Elements of probability. Recurrence relations.


Course OutlineWeek Topics5‐6 Relations and Digraphs

Product sets and partitions. Relations and digraphs. Paths in relations and digraphs. Properties of relations. Equivalence relations.

Data structures for relations and digraphs. Operations on relations. Transitive closure and Warshall’s algorithm.


Course Outline


Week Topics7 Functions

Functions. Functions for computer science. Growth of functions. Permutation functions.

8‐9 Order Relations and Structures

Partially ordered sets. Lattices. Finite Boolean algebras. Functions of Boolean algebras. Circuit design.

Course Outline


Week Topics10 Trees

Trees. Labeled trees. Tree searching. Undirected trees. Minimal spanning trees.

11‐12 Topics in Graph Theory

Graphs. Euler paths and circuits. Transport networks. Matching problems. Coloring graphs.

Course Outline


Week Topics13 Semigroups and Groups

Binary operations. Semigroups. Products and quotients of semigroups. Groups. Products and quotients of groups. Other mathematical structures.

14 Groups and Coding

Coding of binary information and error detection. Decoding and error correction. Public key cryptography.

Important Notes! Attendance is compulsory (University Regulation)

! University dress code

! No mobiles/notes/tabs… (power off or mute mode)

! No late homework will be accepted. No exceptions

! No make‐up exams/quizzes will be given

! Do not be late


INTRODUCTION

What is Discrete Mathematics? Discrete Mathematics is the part of Mathematics devoted to the study of discrete (as opposed to continuous) objects.

Examples of discrete objects: integers, steps taken by a computer program, distinct paths to travel from point A to point B on a map along a road network.

A course in discrete mathematics provides the mathematical background needed for all subsequent courses in computer science.


Discrete Mathematics is a GatewayTopics in discrete mathematics will be important in many courses that you will take in the future:

Computer Architecture, Data Structures and Algorithms, Programming Languages and Compilers, Computer Security, Databases, Artificial Intelligence, Networking, Theory of Computation, …


Problems of Discrete Mathematics How many ways can a password be chosen following specific rules?

How many valid Internet addresses are there?

What is the probability of winning a tournament?

Is there a link between two computers in a network?

How can I identify spam email messages?

How can I encrypt a message so that no unintended recipient can read it?


Problems of Discrete Mathematics How can we build a circuit that adds two integers?

What is the shortest path between two cities using a transportation system?

How can we represent English sentences so that a computer can reason with them?

How can we prove that there are infinitely many prime numbers?

How can a list of integers be sorted so that the integers are in increasing order?


Goals of Discrete Mathematics CourseDiscrete Structures:

Abstract mathematical structures that represent objects and the relationships between them. Examples are sets, strings, sequences, permutations, relations, graphs, trees, and finite state machines.

Combinatorial Analysis:Techniques for counting objects of different kinds.

Mathematical Reasoning:Ability to read, understand, and construct mathematical arguments and proofs.


Goals of Discrete Mathematics CourseAlgorithmic Thinking:

One way to solve many problems is to specify an algorithm.

An algorithm is a sequence of steps that can be followed to solve any instance of a particular problem.

Algorithmic thinking involves specifying algorithms, analyzing the memory and time required by an execution of the algorithm, and verifying that the algorithm will produce the correct answer.


Goals of Discrete Mathematics CourseApplications and Modeling:

It is important to appreciate and understand the wide range of applications of the topics in discrete mathematics and develop the ability to develop new models in various domains.

Concepts from discrete mathematics have not only been used to address problems in computing, but have been applied to solve problems in many areas such as chemistry, biology, linguistics, geography, business, etc.


FUNDAMENTALS

Sets and SubsetsDefinition: A set is any well‐defined collection of objects, called the elements or members of the set.

Examples:

the collection of computers in the Lab;

the collection of students in IIUM.

Well‐defined: it is possible to decide if a given object belongs to the collection or not.

The description of a set: to list the elements of the set between braces:


SetsNotes:

the listing order of the elements is not important:

the repetition of the elements can be ignored:

Notations:

uppercase letters, denote sets

lowercase letters, denote the elements of sets


SetsNotations:

: is an element of .

: is not an element of .

Example:


SetsQ: how to describe a set if it is impossible or inconvenient

to list its elements?

A: define a set by specifying a property that the elements of the set have in common.

“the set of all such that ”

denotes a statement concerning to

Example: ?


Sets

The empty set, denoted by or , has no elements


Empty SetExercise: Which of the following sets are the empty set?

1.

2.

3.

4.


SetsDefinition: Two sets and are equal if they have the same elements, we write .

Example:


SubsetsDefinition: If every element of is also an element of , then we say that is a subset of , and we write .

• Venn diagrams show relationships between sets.

Example: , ,

Example: , ,

Example: , Q: ? ?

• A “universal set” contains all objects for which the discussion is meaningful.


SubsetsDefinition: A set is called finite if it has distinctelements, and is called the cardinality of , and is denoted by .

Definition: A set that is not finite is called infinite.

Definition: The set of all subsets of is called the power set of , and is denoted by or .

Example: Let


SubsetsExercise: Let . Identify each of the following is true or false.

1.

2.

3.

4.


Operations on SetsDefinition: If and are sets, we define their union as the set consisting of all elements that belong to orand denote it by .

Example: Let and .

• Venn diagram?


Operations on SetsDefinition: If and are sets, we define their intersection as the set consisting of all elements that belong to both and and denote it by .

Example: Let and .

Example: Let and .

• Venn diagram?35© S. Turaev, CSC 1700 Discrete Mathematics

Operations on Sets

The union of

The intersection of


Operations on SetsDefinition: If and are sets, we define the complement of w.r.t. (or the difference) as the set consisting of all elements that belong to but not to and denote it by (or ).

Example: Let and .


Operations on SetsDefinition: If is a universal set containing , is called the complement of and is denoted by .

Example: Let and .

• Venn diagram?


Operations on SetsDefinition: If and are sets, we define the symmetric difference as the set consisting of all elements that belong to or to , but not to both and , and denote it by .

Example: Let and .


Algebraic PropertiesCommutative properties:

Associative properties:

Distributive properties:


Algebraic PropertiesIdempotent properties:

Properties of a universal set:

Properties of the empty set:


Algebraic PropertiesProperties of the complement:


Exercise

Let , ,

, ,

and . Compute:

1. 2. 3.

4. 5. 6.

7. 8. 9.


The Addition PrincipleTheorem (addition principle): If and are finite sets, then

Example: Let and

Theorem: If and are finite sets, then


ExerciseIn a survey of 260 college students, the following date were obtained:

• 64 had taken MATH,

• 94 had taken CS,

• 58 had taken IT,

• 28 had taken both MATH and IT,

• 26 had taken both MATH and CS

• 22 had taken both CS and IT

• 14 had taken all three courses

How many students surveyed had taken none of the three courses?


SequencesDefinition: A sequence is a list of objects arranged in a definite order: a first element, a second element, and so on.

If the list stops after steps, then it is finite; if does not stop in any , then it is infinite.

Example:

(finite)

(infinite)


SequencesSequences can be described by formulas:

recursive formula: refers to previous terms to define the next term

explicit formula: describes a term using only its position number.


SequencesExample: define recursive formulas for

Example: write explicit formulas for


introduction fundamentals sets and sequences

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