Welcome to BCT1073

1
Welcome to BCT1073

• Fundamental discrete structure
• BY
• SITI ZANARIAH BT SATARI

2
Synopsis

• This subject introduces and discusses the
  fundamentals of the discrete structures as they
  apply to computer science, focusing on providing
  a basic theoretical foundation for further work.
  Students are exposed to logic, set theory, the
  elementary number theory, functions, relations,
  fundamentals of counting, Boolean algebra, and
  simple proof techniques This course integrates
  symbolic tools, graphical concepts, and numerical
  calculations.

3
Learning Objectives

• Upon the completion of this course, students will
  have the ability to
• Define the basic techniques and methodologies of
  discrete structure.
• Solve any related problem in discrete structure.
• Apply the concepts and methods studied into other
  related courses.

4
Generic Skills

• At the end of the course, the students should
  have acquired the various generic skills targeted
  as below
• Team-working Student able to work in group to
  complete the given assessment in a time given.
• Communication skills Student able to communicate
  effectively in written and oral form through
  group discussion (assignment), tutorial and
  presentation session.
• Problem solving Implement the appropriate skills
  into problem solving process.

5
Teaching Method

• Lecture - 3 hours per week
• Tutorial - 1 hours per week
• Independent Study - 1 credit 40 hours
• (120/15 8 hours per week)
• Individual and Group Assignment - 4 (10)

6
DISCRETE STRUCTURE IS A PART OF discrete
MATHEMATICS
7
THEMES in DISCRETE MATHS

• Mathematical reasoning
• Read, comprehend and construct mathematical
  arguments
• Mathematical logic, proof techniques
• Combinatorial Analysis
• Ability to count or enumerate objects
• Counting techniques
• Discrete Structures
• Abstract mathematical structures used to
  represent discrete objects and relationships
  between these objects
• Sets, permutation, relations, graphs, trees and
  finite state machine
• Algorithmic Thinking
• Described an algorithm and then implement by
  computer program (coding)
• Applications and Modeling
• Many applications in Computer Science and Data
  Networking
• New model can be developed

8
What is discrete MATHEMATICS???
9
DISCRETE MATHS

• The part of mathematics devoted to the study of
  discrete objects
• Discrete means consisting of distinct or
  unconnected elements or separable parts.
  (Opposite of continuous.)
• Used when
• objects are counted
• relationship between finite (or countable) sets
  are studied
• processes involving a finite number of steps are
  analyzed

Discrete Mathematics - The study of discrete,
mathematical objects and structures.
10
Some Problems involving Discrete Mathematics

• How many ways are there to chose a valid password
  on a computer system?
• Is there a link between two computers in a
  network?
• What is the shortest path between two cities
  using a transportation system?
• How can a circuit that ads two integers be
  design?
• How many valid Internet addresses are there?

You will learn the discrete structures and
techniques need to solve these problems in
BCT1073 and BCT2083 course
11
Why study DISCRETE MATHS ?

• You can develop your mathematical maturity, that
  is your ability to understand and create
  mathematical arguments.
• You should be able to
• Learn a particular set of mathematical facts
• Learn how to apply them
• Think mathematically
• DM is the gateway to more advanced course in
  mathematical sciences
• DM provides the mathematical foundations for many
  computer science course including data
  structures, algorithms, database theory, automata
  theory, formal languages, compiler theory,
  computer security and operating systems
• DM contains the necessary mathematical background
  for solving many application problems.

12
What is DISCRETE STRUCTURE?

• Discrete (? discreet!) - Composed of
  distinct, separable parts. (Opposite of
  continuous.)
• Structures - Objects built up from simpler
  objects according to some definite pattern.

13
What is fundamental discrete structure???
14
Why DISCRETE STRUCTURE?

• Mathematical reasoning
• Read, comprehend and construct mathematical
  arguments
• Mathematical logic, proof techniques
• Combinatorial Analysis
• Ability to count or enumerate objects
• Counting techniques
• Discrete Structures
• Abstract mathematical structures used to
  represent discrete objects and relationships
  between these objects
• Sets, permutation, relations, graphs, trees and
  finite state machine
• Algorithmic Thinking
• Described an algorithm and then implement by
  computer program (coding)
• Applications and Modeling
• Many applications in Computer Science and Data
  Networking
• New model can be developed

Only these themes will be discussed further in
BCT1073
15
Some Discrete Structures

• Propositions
• Predicates
• Proofs
• Sets
• Functions
• Orders of Growth
• Algorithms
• Integers
• Summations

• Sequences
• Strings
• Permutations
• Combinations
• Relations
• Graphs
• Trees
• Logic Circuits
• Automata

16
Relationships Between Structures

• ? ? Can be defined in terms of

17
Some Notations Well Learn

18
Topics covered
19
1 LOGIC

• Proposition
• Logical Connectives
• Logical Equivalences
• Predicates and Quantifiers
• Modus Ponens and Modus Tollens (Valid Invalid
  Arguments)

20
2 SET THEORY

• Set Terminologies and Concepts
• Operation on Sets
• Cartesian Products
• Power Sets
• Applications of Set Theory

21
3 ELEMENTARY NUMBER THEORY

• Factorability
• Primes
• The Division Algorithm
• Greatest Common Divisors (GCD)
• Least Common Multiples (LCM)
• Euclidean Algorithm
• Modular Arithmetic
• The Chinese Remainder Theorem

22
4 FUNCTIONS AND RELATIONS

• Introduction to Functions
• One-to-one Functions
• Onto Functions
• Relations and Their Properties
• Relations
• Reflexive
• Symmetric
• Transitive

23
5 BASIC OF COUNTING

• Counting Arguments
• More Complex Counting Problems
• The Inclusion-Exclusion Principle
• Tree Diagrams
• The Pigeonhole Principle
• Permutations
• Combinations
• Binomial Coefficients

24
6 BOOLEAN ALGEBRA

• Boolean Values
• Boolean Expression and Boolean Functions
• Identities of Boolean Algebra
• Abstract Boolean Algebra
• Representing Boolean Functions
• Logic Gates
• Minimization of Circuits

25
7 PROOF TECHNIQUES

• Introduction
• Direct Proofs
• Indirect Method
• Contradiction Method
• Mathematical Induction

26
ASSESSMENT

• Item Percentage ()
• Quizzes (5) 10
• Assignment (4) 10
• Test 1 20
• Test 2 20
• Final Exam 40
• Total 100

27
REFERENCES

• Rosen K.H., Discrete Mathematics Its
  Applications, (Seventh Edition), McGraw-Hill,
  2007.
• Kolman B., Busby R.C. Ross S.C., Discrete
  Mathematical Structures. (Fifth Edition),
  Prentice-Hall, 2004.
• Epp S.S, Discrete Mathematics with Applications,
  (Third Edition), Thomson Learning, 2004.
• Gossett E., Discrete Mathematics with Proof,
  Prentice-Hall, 2003.
• Malik D.S. Sen M.K., Discrete Mathematical
  Structures, Thomson Learning, 2004.
• http//www.mhhe.com/rosen

28
NOTES TUTORIAL

• Please download and print out all notes and
  tutorials from each chapter before class.
• http//notes.ump.edu.my OR e-learn
• Tutorial class will be handled by Mrs Santi Wulan

29
CLASS RULES

• Always come early to my Class
• Class will start on time
• Class will only start after DOA
• Everyone must bring a copy of power point slide
  (notes) and calculator
• Please show good attitude
• Give me a copy of MC letter or any letter approve
  by your PA if you fail to come to my class. If
  not you will get KT for your softskill.

30
LAST word

• Please enjoy your class.
• If you have any problem in Mathematics feel free
  to visit me at my room
• First floor FSKKP
• My phone number
• 09-5492145 (office)
• 019-7699240

31
MOTIVATION

• I hear... and I forget
• I see... and I remember
• I do... and I understand
• Ancient Chinese Proverb

32
MOTIVATION
"One cannot change yesterday. But only make the
most of today. Look with hope toward tomorrow."
So do your best and get A in this course !!!