dicrete structure

1. MA - 210
Discrete Structures
Introduction
Lecture-1

2. Overview of this Lecture
• Course Administration
• What is it about?
• Course Themes, Goals, and Syllabus
• Propositional logic

3. Course Administration

4. Timings
Lecture Hours:
Monday (α ): 12:00 PM – 03:00 PM
Tuesday (Ω): 12:00 PM – 03:00 PM
Location:
Class Room 5
Lecturer: Mona Waseem
Office: Faculty Office (Room-10), Ground Floor, CPED
Phone: 051-9047
Email: mona.waseem@uettaxila.edu.pk

5. Assessments
Homework (12%)
Quiz (13%)
Midterm (25%)
Final (50%)

6. Homework and Quiz
• Homework is very important. It is the best way for you to
learn the material.
• You can discuss the problems with your classmates, but all
work handed in should be original, written by you in
your own words.
• Homework should be handed in in class.
• 20% penalty for each day late.
• Most Probably Quiz will be announced a week
before BUT there will be NO Re-Take for Quiz
in any Case.

7. Textbook
Discrete Mathematics and Its Applications
by Kenneth H. Rosen, 7th Edition

8. Reference Book (1)
Discrete Mathematics
by Richard Johnsonbaugh
7th or 8th Edition

9. Reference Book (2)
Discrete Mathematics
by Edgar Goodaire and Michael Parmenter
3rd Edition.

10. Course Learning Outcome (CLO’s)
S. No. Course Learning Outcome (CLO’s)
Bloom’s
Learning
Level
1.
Recall the basic concepts of logic and proofs, set theory,
relations in functions and complexity of algorithms.
C1, PLO-1
2.
Demonstrate the fundamental concepts of counting, graphs,
trees, and Mathematical Induction.
C2, PLO-1
3.
Confidently apply the knowledge learnt to solve
mathematical problems in computer science and engineering
disciplines.
C3, PLO-2
Learning Levels (LL): Knowledge (LL1), Comprehension (LL2), Application
(LL3), Analysis (LL4), Synthesis (LL5), Evaluation (LL6)

11. Overview of this Lecture
Course Administration
What is MA-210 about?
Course Themes, Goals, and Syllabus

12. Continuous vs. Discrete Math
Why is it computer science/engineering?
Mathematical techniques for DM
What is MA-210 about?

13. What Are Discrete Structures?
Discrete math is mathematics that deals with discrete objects:
• Discrete Objects are those which are separated from (distinct from)
each other, such as integers, rational numbers, houses, people, etc.
Real numbers are not discrete.
• In this course, we’ll be concerned with objects such as integers,
propositions, sets, relations and functions, which are all discrete.
• We’ll learn concepts associated with them, their properties, and
relationships among them.
Discrete structures are:
– Theoretical basis of computer science
– A mathematical foundation that makes you think logically
– A prerequisite course to understand the fundamentals of programming

14. Discrete vs. Continuous Mathematics
Continuous Mathematics
It considers objects that vary continuously;
Example: analog wristwatch (separate hour, minute, and second hands).
From an analog watch perspective, between 1 :25 p.m. and 1 :26 p.m.
there are infinitely many possible different times as the second hand moves
around the watch face.
Real-number system --- core of continuous mathematics;
Continuous mathematics --- models and tools for analyzing real-world
phenomena that change smoothly over time. (Differential equations etc.)

15. Discrete vs. Continuous Mathematics
Discrete Mathematics
It considers objects that vary in a discrete way.
Example: digital wristwatch.
On a digital watch, there are only finitely many possible different times
between 1 :25 P.m. and 1:27 P.m. A digital watch does not show split
seconds: no time between 1 :25:03 and 1 :25:04. The watch moves from one
time to the next.
Integers --- core of discrete mathematics
Discrete mathematics --- models and tools for analyzing real-world
phenomena that change discretely over time and therefore ideal for studying
computer science – computers are digital! (numbers as finite bit strings; data
structures, all discrete! Historical aside: earliest computers were analogue.)

16. Discrete vs Continuous

17. Why is it computer science/engineering?
(examples)
What is MA-210 about?

18. • Computers use discrete structures to represent
and manipulate data.
• is the basic building block for becoming a
Computer Scientist
• Computer Science is not Programming
• Computer Science is not Software Engineering
• Edsger Dijkstra: “Computer Science is no more
about computers than Astronomy is about
telescopes.”
• Computer Science is about problem solving.
Why Discrete Mathematics? (I)

19. • Mathematics is at the heart of problem solving
• Defining a problem requires mathematical rigor
• Use and analysis of models, data structures,
algorithms requires a solid foundation of
mathematics
• To justify why a particular way of solving a
problem is correct or efficient (i.e., better than
another way) requires analysis with a well-
defined mathematical model.
Why Discrete Mathematics? (II)

20. • Your boss is not going to ask you to solve
– an MST (Minimal Spanning Tree) or
– a TSP (Travelling Salesperson Problem)
• Rarely will you encounter a problem in an
abstract setting
• However, he/she may ask you to build a rotation
of the company’s delivery trucks to minimize
fuel usage
• It is up to you to determine
– a proper model for representing the problem and
– a correct or efficient algorithm for solving it
Problem Solving requires mathematical rigor
(Accuracy, Objectivity)

21. Discrete Mathematics in the Real
World
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22.  Computers run software and store files.
 The software/files both stored as huge strings of 1s and 0s.
 Binary math is discrete mathematics.
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25. Number Theory:
RSA and Public-key Cryptography
Alice and Bob have never met but they would like to
exchange a message. Eve would like to eavesdrop.
They could come up with a good
encryption algorithm and exchange the
encryption key – but how to do it without
Eve getting it? (If Eve gets it, all security
is lost.)
CS folks found the solution:
public key encryption. Quite remarkable that is feasible.
E.g. between you and the Bank of America.

26. Number Theory:
Public Key Encryption
RSA – Public Key Cryptosystem (why RSA?)
Uses modular arithmetic and large primes  Its security comes from the computational difficulty
of factoring large numbers.

27. RSA Approach
• Encode:
• C = Me (mod n)
• M is the plaintext; C is ciphertext
• n = pq with p and q large primes (e.g. 200 digits long!)
• e is relative prime to (p-1)(q-1)
• Decode:
• Cd = M (mod pq)
• d is inverse of e modulo (p-1)(q-1)
The process of encrypting and decrypting a message
correctly results in the original message (and it’s fast!)
Hmm??
What does this all mean??
How does this actually work?
Not to worry. We’ll find out.

28. Online Shopping
 Encryption and decryption are part of
cryptography, which is part of discrete
mathematics. For example, secure internet
shopping uses public-key cryptography.
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29. Analog Clock
 An Analog Clock has gears inside, and the sizes/teeth
needed for correct timekeeping are determined using
discrete math of modular arithmetic.
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30. Dijkstra’s Algorithm and Google Maps
 Google Maps uses discrete mathematics to
determine fastest driving routes and times.
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31. Railway Planning
 Railway planning uses discrete math: deciding how to
expand train rail lines, train timetable scheduling, and
scheduling crews and equipment for train trips use
both graph theory and linear algebra.
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33. Computer Graphics
 Computer graphics (such as in video games) use linear
algebra in order to transform (move, scale, change
perspective) objects. That's true for both applications
like game development, and for operating systems.
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34. Smarter Phones for All
 Cell Phone Communications: Making efficient use of the
broadcast spectrum for mobile phones uses linear
algebra and information theory. Assigning frequencies
so that there is no interference with nearby phones can
use graph theory or can use discrete optimization.
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35. Graphs and Networks
Many problems can be
represented by a graphical network
representation.
•Examples:
– Distribution problems
– Routing problems
– Maximum flow problems
– Designing computer / phone / road networks
– Equipment replacement
– And of course the Internet
Aside: finding the right
problem representation
is one of the key issues
in this course.

36. Sub-Category Graph
No Threshold
New Science of Networks
NYS Electric
Power Grid
(Thorp,Strogatz,Watts)
Cyber communities
(Automatically discovered)
Kleinberg et al
Network of computer scientists
ReferralWeb System
(Kautz and Selman)
Neural network of the
nematode worm C- elegans
(Strogatz, Watts)
Networks are
pervasive
Utility Patent network
1972-1999
(3 Million patents)
Gomes,Hopcroft,Lesser,Selman

37. Applications of Networks
Applications
Physical analog
of nodes
Physical analog
of arcs
Flow
Communication
systems
phone exchanges,
computers,
transmission
facilities, satellites
Cables, fiber optic
links, microwave
relay links
Voice messages,
Data,
Video transmissions
Hydraulic systems
Pumping stations
Reservoirs, Lakes
Pipelines
Water, Gas, Oil,
Hydraulic fluids
Integrated
computer circuits
Gates, registers,
processors
Wires Electrical current
Mechanical systems Joints
Rods, Beams,
Springs
Heat, Energy
Transportation
systems
Intersections,
Airports,
Rail yards
Highways,
Airline routes
Railbeds
Passengers,
freight,
vehicles,
operators

38. Finding Friends on Facebook
 Graph theory can be used in speeding up Facebook
performance.
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39. DNA Fragment Assembly
 Graph theory is used in DNA sequencing.
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40. Applications Of Discrete Structures
Algorithms can be used in searching for a letter or number in the list.
The study of these sorting algorithms often involves analyzing their time
complexity, space complexity, and other performance characteristics
using discrete mathematics concepts.

41. Applications Of Discrete Structures
 Set theory can be used to determine number of students enrolled in
discrete structures and number of students enrolled in applied
physics in this semester.

42. Learning Discrete Mathematics make you
a better Programmer?
 One of the most important competences – that one must
have; as a program developer is to be able to choose the right
algorithms and data structures for the problem that the
program is supposed to solve.
 The importance of discrete mathematics lies in its central role
in the analysis of algorithms and in the fact that many
common data structures – and in particular graphs, trees, sets
and ordered sets – and their associated algorithms come from
the domain of discrete mathematics.
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43. Logic and Proofs
 Programmers use logic. All the time. While everyone
has to think about the solution, a formal study of
logical thinking helps you organize your thought
process more efficiently.
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44. Logic:
Hardware and software specifications
One-bit Full Adder with
Carry-In and Carry-Out
4-bit full adder
Example 2: System Specification:
–The router can send packets to the edge system only if it supports the new address space.
–For the router to support the new address space it’s necessary that the latest software release be installed.
–The router can send packets to the edge system if the latest software release is installed.
–The router does not support the new address space.
Example 1: Adder
How to write these specifications in a formal way? Use Logic.
Formal: Input_wire_A
value in {0, 1}

45. Propositional Logic
 System specifications demonstrate the requirements of
a system. These requirements are stated in English
sentences which can be ambiguous. Therefore,
translating these English requirements into logical
expressions removes the ambiguity and check for the
consistency of specifications.
 Compound propositions and quantifiers can be used to
express the specifications and then find an assignment
of truth values that make all the specifications true.

46. Why Proofs?
 Why proofs? The analysis of an algorithm requires
one to carry out (or at the very least be able to
sketch) a proof of the correctness of the algorithm
and a proof of its complexity
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47. Moral of the Story???
 Discrete Math is needed to see mathematical
structures in the object you work with, and
understand their properties. This ability is important
for computer/software engineers, data scientists,
security and financial analysts.
 it is not a coincidence that math puzzles are often
used for interviews.
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48. Discrete Mathematics is a Gateway
Course
 Topics in discrete mathematics will be important in
many courses that you will take in the future:
 Computer Science: Computer Architecture, Data
Structures, Algorithms, Programming Languages,
Compilers, Computer Security, Databases, Artificial
Intelligence, Networking, Graphics, Game Design, Theory
of Computation, Game Theory, Network Optimization ……
 Other Disciplines: You may find concepts learned here
useful in courses in philosophy, economics, linguistics,
and other departments.
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49. Course Themes, Goals, and Course Outline

50. Goals of MA-210
Introduce students to a range of mathematical tools from discrete
mathematics that are key in computer science
Mathematical Sophistication
How to write statements rigorously
How to read and write theorems, lemmas, etc.
How to write rigorous proofs
Areas we will cover:
Logic and proofs
Set Theory
Number Theory
Counting and combinatorics
Practice works!
Actually, only practice works!
Note: Learning to do proofs from
watching the slides is like trying to
learn to play tennis from watching
it on TV! So, do exercises!

51. Tentative Topics MA-210
Logic and Methods of Proof
Propositional Logic --- SAT as an encoding language!
Predicates and Quantifiers
Methods of Proofs
Number Theory
Modular arithmetic
RSA cryptosystems
Sets
Sets and Set operations
Functions
Counting
Basics of counting
Pigeonhole principle
Permutations and Combinations

52. Topics MA-210
Graphs and Trees
Graph terminology
Example of graph problems and algorithms:
graph coloring
TSP
shortest path
Min. spanning tree

53. Bart Selman
CS2800
What’s your job?
• Build a mathematical model for each scenario
• Develop an algorithm for solving each task
• Prove that your solutions work
– Termination: Prove that your algorithms terminate
– Completeness: Prove that your algorithms find a
solution when there is one.
– Soundness: Prove that the solution of your algorithms is
always correct
– Optimality (of the solution): Prove that your algorithms
find the best solution (i.e., maximize profit)
– Efficiency, time & space complexity: Prove that your
algorithms finish before the end of life on earth

54. Bart Selman
CS2800
Discrete Structures
Logic

55. Bart Selman
CS2800
Logic
• Logic, is the study of reasoning and making
sense of things in a systematic and clear way.
• It helps us figure out if our ideas and
arguments make sense, and it provides a
way to think through problems and come to
reliable conclusions.
• Logic is like having a set of steps to follow to
make good decisions and understand things
better.

56. Bart Selman
CS2800

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CS2800

58. Bart Selman
CS2800
Activity
Which of these sentences are propositions? What are the
truth values of those that are propositions?
• Boston is the capital of Massachusetts.
• Miami is the capital of Florida.
• 2+3=5
• x+7=10
Answer this question

59. Bart Selman
CS2800
Answers
1. Boston is the capital of Massachusetts.
1. Proposition: Yes
2. Truth Value: True
2. Miami is the capital of Florida.
1. Proposition: Yes
2. Truth Value: False (Miami is not the capital of Florida; it is
Tallahassee)
3. 2 + 3 = 5.
1. Proposition: Yes
2. Truth Value: True (mathematically correct)
4. x + 7 = 10.
1. Not a Proposition (contains a variable; its truth depends on the
value of x)

60. Bart Selman
CS2800

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CS2800

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CS2800

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CS2800

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CS2800

65. Bart Selman
CS2800
Activity
What is the negation of these propositions
• Mei has an MP3 player.
• There is no pollution.
• 2+1=3
• The summer in Maine is hot and sunny.

66. Bart Selman
CS2800
Answers
1. Mei has an MP3 player:
1. Negation: Mei does not have an MP3 player.
2. Symbolically: ¬p
2. There is no pollution:
1. Negation: There is pollution.
2. Symbolically: ¬q
3. 2+1=3:
1. Negation: 2+1 is not equal to 3.
2. Symbolically: ¬(2+1=3)
4. The summer in Maine is hot and sunny:
1. Negation: The summer in Maine is not hot and sunny (it could be
cold or not sunny).
2. Symbolically: ¬(p∧q)

67. Bart Selman
CS2800

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CS2800

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CS2800

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CS2800

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CS2800

72. Bart Selman
CS2800
Activity
Let p and be the propositions
p: It is below freezing
q: It is snowing
Write these propositions using p and q and logical
connectives
a) It is below freezing and snowing
b) It is below freezing but not snowing
c) It is not below freezing and it is not snowing
d) It is either snowing or below freezing(or both)

73. Bart Selman
CS2800
Answers
a. It is below freezing and snowing:
p∧q
b. It is below freezing but not snowing:
p∧¬q
c. It is not below freezing and it is not
snowing:
¬p∧¬q
d. It is either snowing or below freezing (or
both):
p∨q

74. Bart Selman
CS2800

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CS2800

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CS2800

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CS2800

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CS2800

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CS2800

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CS2800

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CS2800

83. Bart Selman
CS2800
Activity
Let p and be the propositions
p: It is below freezing
q: It is snowing
Write these propositions using p and q and logical
connectives
a) If it is below freezing, then it is snowing.
b) It is snowing whenever it is below freezing.
c) That it is below freezing is necessary and sufficient for it to be
snowing

84. Bart Selman
CS2800
Activity
Let p and be the propositions
p: It is below freezing
q: It is snowing
Write these propositions using p and q and logical
connectives
a) If it is below freezing, then it is snowing.
p→q
b) It is snowing whenever it is below freezing.
p→q
c) That it is below freezing is necessary and sufficient for it to be
snowing
p↔q

85. Bart Selman
CS2800

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CS2800

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CS2800

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CS2800

92. Bart Selman
CS2800
Construct a truth table for each of these compound
propositions
• p ∧ ¬p
• p v ¬p
• p ⊕ (p v q)
• (p v q) → (p ∧ q)
• p ⊕ ¬ q
Activity

93. Bart Selman
CS2800
The end