Discrete Mathematics

Lt Col Muhammad Saleem(Retd)

Military College of SignalsNational University of Sciences and Technology, Pakistan

Fall, 2015

saleem@mcs.edu.pk

Agenda

Course DetailsObjectivesGrading Policy

Introduction to Discrete MathCredit hrs 3+0Contact hrs 7+0 (Incl 1xSTI)Pre Requisites Nil

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Resources

Lecture Slides will be available on LMS

Quizzes and Assignments will be loaded on LMS.

Additional references shall be provided (if any)

Text BookDiscrete Mathematics with Applications (Min Third Edition)

by Susanna S. Epp. The Latest edition aval in marketReference

Discrete Mathematics and Its Applications (fourth edition) by Kenneth H. Rosen

Discrete Mathematics by Ross and Wright

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Grading PolicyAssignments (3-5) 10%Quizzes (3-5)

10%OHT-1 15%OHT-2 15%End-term 50%

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AssignmentsQuizzes

Mostly unannouncedOccasionally

announced

Point System - every activity in the class gives you a certain number of points

Assignment Policies

No late work will be accepted (Unless arrangements have been made in

advanceBy the C/Snr)

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Policies – AttendanceDeficiency in attendance may lead to

termination or relegation or F-Grade

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No re-take of quizzes in any case except that the leave is officially approved by department

Policies

Course Senior:

Make list of e-mails of all students

• It is preferred to contact in office hours.

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I reserve the right to be more lenient

Course ObjectivesPreparation for the mathematical aspects of

other computer science courses

Learn the basic results in logic, Relations, Set, Trees

and graph theory.

Cultivate a sense of familiarity and ease in working with mathematical notation and common concepts in discrete mathematics.

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What is Discrete Mathematics?

:

Discrete Mathematics concerns processes that consist of a sequence of individual steps.

Topics in Discrete MathSets and Logic (Useful in digital circuits & hardware

design)Relations (Relational DBs)Graphs Theory (describe problems in Data bases,

Operating Systems and Networks) Trees (Data structure)Number Theory (use in crypto-systems)Recursion (Breaking down the problems)Complexity Theory (Numerical solution of differential

eqns)Functions (Mathematical modelling)Counting and Probability (Coding theory)

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Course outlineLogic: logical form, logical Equivalance, Conditional statements. Valid&Invalid arguments, Predicate and quantifiersRelations: Relation properties, n-ary relations and Applications , Closures of Relations and equivalence relation. Graphs: Introduction to Graphs, isomorphism, Connectivity, Euler and Hamilton Paths, Shortest Path Problems, planner Graphs and coloringTrees: Introduction and Applications, Tree traversal, sorting

Hints for success

Read the textbook.

Lectures really do help!

Set up study groups and attend sections.

Do the homework

Analog vs Digital Analog means continuous

Analog parameters have continuous range of

values

Example: Temperature is an analog parameter

Temperature increases/decreases continuously

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100oC

timetem

pera

ture

Kettle removed from stove

Analog vs Digital

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Analog

Digital

Analog vs Digital Digital signals are non-continuous i.e. discrete

Consist of fixed set of digits. E.g. number of months in a year = 12; digits = {1,2,3,….,10,11,12} note that 11.3 or 4.9 are invalid here

Abrupt transition (jumping) from one digit to another

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Val

ue

1 23 4 5 6 7 8 9 10 11 12

Propositional Logic

When we talk about Philosophy & Logic the first name that comes into our mind is that of Aristotle (384 BC- 322 BC) who followed a comprehensive system of ideas about human nature and the nature of the reality we live in.

Logic , is concerned with method of reasoning. Aristotle, the 1st who studied logical reasoning, logic provides the basis for many areas of CS (DLD , Al and Automata theory )

04/18/2023

Propositional LogicWhat’s a proposition?

Propositions Not Propositions

3 + 2 = 32 Bring me coffee!

DM is Aysha favorite class.

CS173 is her favorite class.

Every cat has 4 legs. 3 + 2

There is other life in the universe.

Do you like Cake?

A proposition is a declarative statement that’s either TRUE or FALSE (but not both).

Propositional Logic

TerminologiesInductive LogicDeductive LogicAxiomsPostulatesPrimitive propositionArgumentSentenceStatementCompound statement

Symbolic Logical /Connectors

AND vOR ^NOT -- Implies Equivalent to

Precedence for Logical Connectors

Negation has precedence over AND &ORa. Inner bracketb. Negationc. Conjunction d. Disjunction (c&d are at equal

level)e. implication

Analysis of complex statement

Constructing Truth Tablea. Number of Rowsb. Number of Colum's