K L University

Department of Computer Science & Engineering

Course Handout for Discrete Mathematics II Year B.Tech PROGRAM

A.Y.2015-16, II Semester

Course Name : Discrete Mathematics

Course Code : 13-BS 206

L-T-P structure : 3-0-0

Course Credits : 3

Course Coordinator : Dr.C.M.Sheela Rani

Course Instructors : Dr.K.Subramanyam, Dr.S.Satyanarayana, Mr.V.Naga Bhushanam, Ms.T.Siva Priya,

Mrs.T.Sajana, Mrs. J.Yamini Devi, Ms. Yamini Gupta, Mr.U.Narayankar.

Course Teaching Associates:

Course Objective: The objective of this course is to enable the student to equip with discrete mathematical structures necessary to understand optimization techniques while working with shortest path problems, routing problems, maximum-flow problems. It introduces logical thinking skills which enables them to solve real time problems. It provides the mathematical foundations required for designing the circuits.

Course Rationale: This course covers discrete mathematics which emphasizes mathematical definitions and proofs as well as applicable methods. Topics include Sets, Relations, methods of proofs, Induction; Counting principles, Permutations and Combinations; Elementary Graph theory, Lattices; Recurrence relations and Generating functions. Student will be able to use these techniques in subsequent courses like Design and Analysis of Algorithms, Cryptography & Network security, Parallel Processing, Natural Language Processing, Machine Learning, Software Engineering.

Course Outcomes (CO):

CO No: Course Outcomes SO BTL

1 Apply various Set Operations and Logical Inferences for solving problems and the principle of Mathematical Induction. a, e 2

2Analyze Combinatorial and Permute Analysis, Binomial theorem, Multinomial theorem and Principle of Inclusion and Exclusion. e 2

3 Analyze different types of Graphs, Lattices, Sorting and Searching techniques and Applications of Graphs a 2

4Apply procedure for solving Spanning Trees and different methods for solving Recurrence Relations.

e 2

COURSE OUTCOME INDICATORS (COI):

CO No. COI-1 COI-2 COI-3

1Apply various operations on sets to solve problems

Understand Relations and Functions, Method of proofs

Apply the rules of Logical Inferences and steps of Mathematical Induction.

2Analyze basics of Counting and solves problems related to Permutations and Combinations.

Classify different types of Enumerations in Permutations and Combinations.

Use Binomial Theorem, Multinomial Theorem and Principle of Inclusion and Exclusion for solving related problems.

3Use the properties of Relations for solving problems on Lattices.

Analyze Graphs and Sub graphs, Sorting and Searching techniques

Applications of Graphs :Euler’s Formula, Hamiltonian,Four-Color problem.

4Apply DFS, BFS, Prim’s, Kruskal’s algorithms.

Apply the procedure for solving recurrence relations using Generating functions.

Illustrate the procedure for solving recurrence relations and solves related problems.

SYLLABUS (As approved by BOS):

SYLLABUS:

Foundations: Basics, Sets and Operations of Sets, Relations and Functions, Some methods of Proofs and Problem Solving Strategies, Fundamentals of Logic, Logical Inferences, Methods of Proof of an Implication, First order logic and Other methods of Proof, Rules of Inference for Quantified Propositions, Mathematical Induction.

Elementary Combinatory: Basics of Counting, Combinations and Permutations, Enumeration of Combinations and Permutations, Enumerating Combinations and Permutations with repetitions, Enumerating Permutations with constrained repetitions, Binomial Coefficients, The binomial and multinomial theorems, The principle of inclusion - exclusion.

Recurrence Relations: Generating Functions of Sequences, Calculating Coefficients of Generating Functions, Recurrence Relations, Solving Recurrence relations by Substituting and Generating Functions, The Method of Characteristic Roots, Solution of Inhomogeneous Recurrence Relations.

Relations and Digraphs: Relations and Directed Graphs, Special Properties of Binary Relations, Equivalence Relations, Ordering Relations, Lattices, and Enumerations, Operations on Relations, Paths and Closures, Directed Graphs and Adjacency Matrices,

Applications: Sorting and Searching, Topological Sorting. Graphs: Basic Concepts, Isomorphism’s and Sub graphs, trees and their Properties, Spanning Trees, Directed Trees, Binary trees, Planar Graphs, Euler’s Formula, Multi graphs and Euler Circuits, Hamiltonian Graphs, Chromatic Numbers, The Four-Color Problem.

TEXT BOOKS:

1. Joe L. Mott, Abraham Kandel, Theodare P.Baker “Discrete mathematics for Computer Scientists and mathematicians” 2007, Second Edition, PHI,.

REFERENCES:

1. Kenneth H Rosen, “Discrete Mathematics and its Applications”, 2007,Tata McGraw Hill Publishing Company Limited, New Delhi, Sixth Edition,.

2. Tremblay J P and Manohar R, “Discrete Mathematical Structures with Applications to Computer Science”, 2007, Tata McGraw

Hill Publishing Company Limited, New Delhi.

COURSE DELIVERY PLAN:

Sess. No. CO COI Topic (s) Teaching-Learning

MethodsEvaluation

Components

1 1 1 Introduction to Discrete mathematics Marker and Talk, PPT - Observe

Test-1, End Exam

2 1 1 Sets , operations on sets

Marker and Talk, PPT, Programming through Laptop - Observe, Practice

Test-1, End Exam

3 1 2 Relations and Functions

Marker and Talk, PPT, Programming through Laptop - Observe, Practice

Test-1, End Exam

4 1 2 Problem solving strategies , Fundamentals of logicMarker and Talk,

PPT - Observe, Practice

Test-1, End Exam

5 1 3 Logical InferencesMarker and Talk,

PPT - Observe, Practice

Test-1, End Exam

6 1 3 Quantified PropositionsMarker and Talk,

PPT – Observe, Practice

Test-1, End Exam

7 1 3 Mathematical Induction Marker and Talk - Listen, Practice

Test-1, End Exam

8 2 1 Basics of Counting , PermutationsMarker and Talk,

PPT - Observe, Practice

Test-2, End Exam

9 2 1 Combinations, Enumerations without RepetitionsMarker and Talk,

PPT - Observe, Practice

Test-2, End Exam

10 2 2 Enumerations with RepetitionsMarker and Talk,

PPT - Observe, Practice

Test-2, End Exam

11 2 2 Enumerations with Constrained RepetitionsMarker and Talk,

PPT - Observe, Practice

Test-2, End Exam

12 2 3 Binomial Coefficient , Binomial Theorem and Multinomial Theorem Marker and Talk - Listen, Practice

Test-2, End Exam

13 2 3 Principle of Inclusion and Exclusion Marker and Talk - Listen, Practice

Test-2, End Exam

14 3 1 Relations , Properties of binary relations , equivalence and ordering relations

Marker and Talk - Observe, Practice

Test-3, End Exam

15 3 1 Operations on relations , Lattices Marker and Talk - Listen, Practice

Test-3, End Exam

16 3 1 Problems on Lattices Marker and Talk - Listen, Practice

Test-3, End Exam

17 3 2 Graphs – Digraphs , Adjacency matrices ,Isomorphism

Marker and Talk, PPT, Programming through Laptop - Observe, Practice

Test-3, End Exam

18 3 3 Planar graphs, multi graphs, Euler’s formula, Euler’s circuit, Sorting and Searching techniques.

Marker and Talk, PPT, Programming through Laptop - Observe, Practice

Test-3, End Exam

19 3 3 Hamiltonian graphs, Chromatic Numbers, The Four-Color Problem Marker and Talk, PPT - Practice

Test-3, End Exam

20 4 1 Trees – Properties, Directed Trees, Rooted trees, Spanning Trees – DFS, BFS

Marker and Talk, PPT, Programming through Laptop - Observe, Practice

End Exam

21 4 1 Spanning Trees – Prim’s and Kruskal’s algorithms Marker and Talk, PPT - Practice End Exam

22 4 2 Generating functions for sequences Marker and Talk - Listen, Practice End Exam

23 4 3 Calculating coefficients for generating functions, RR – Substitution method

Marker and Talk - Listen, Practice End Exam

24 4 3 RR- Generating functions Marker and Talk - Listen, Practice End Exam

25 4 3 RR-Characteristic Roots Marker and Talk - Listen, Practice End Exam

Session wise Teaching – Learning Plan

Session Number: 1 Session Outcome: Students will be able to understand the need of Discrete Mathematics and its real time applications.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance

10 Introduction of the Faculty, Overview of the Course, Evaluation Scheme 1 Talk - Discussion

30 Overview of the Course Outcomes (Handout) 1 PPT’s - Observe10 Discussion 1 Talk - Participate30 Introduction to Discrete Mathematics 1 PPT’s - Observe10 Discussion 1 Talk - Participate05 Conclusion & Summary Discussion

Session Number: 2Session Outcome: Students will be able to understand what a Set is and its operations.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap of Introduction 1 Talk - Discussion30 Introduction to sets 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One minute paper30 Operations on sets 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 3Session Outcome: Students will be able to understand Relation and Functions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Operations on sets 1 Talk - Discussion30 Relations 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Fish Bowl30 Functions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Quiz05 Conclusion & Summary Discussion

Session Number: 4Session Outcome: Students will be able to analyze different Problem solving strategies and understand the Fundamentals of logic.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Relations and Functions 1 Talk - Discussion30 Problem solving strategies 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Immediate Feedback30 Fundamentals of logic 1 Marker and Talk, PPT’s - Observe10 Active Learning Program 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 5Session Outcome: Students will be able to understand the rules of Logical inferences and solve them.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on fundamentals of Logic 1 Talk - Discussion30 Rules of Logical Inferences 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One minute paper / Chart30 Solving problems related to Logical inferences 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 6Session Outcome: Students will be able to understand the Quantified propositions and solve them.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Logical Inferences 1 Talk - Discussion30 Quantified propositions – formulas 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Immediate Feedback30 Solving problems related to Quantified propositions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program 2 Statement – Opinion - Summary05 Conclusion & Summary Discussion

Session Number: 7

Session Outcome: Students will be able to understand the principle of Mathematical Induction.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Quantified propositions 1 Talk - Discussion30 Steps in Mathematical Induction 1 Marker and Talk - Listen10 Active Learning Program: 2 One minute paper30 Solving problems using the principle of Mathematical Induction 1 Marker and Talk - Listen10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 8Session Outcome: Students will be able to understand the principles of Basic counting, sum rule, product rule and the concept of Permutations.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Course Outcome -1 1 Talk - Discussion30 Basics of counting 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 1 Immediate Feedback30 Permutations 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Statement – Opinion - Summary05 Conclusion & Summary Discussion

Session Number: 9Session Outcome: Students will be able to learn the various methods of Combinations and understand permutations and combinations without repetitions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Permutations 1 Talk - Discussion30 Combinations 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper30 Enumerations without Repetitions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 10Session Outcome: Students will be able to understand different methods of permutations and combinations with repetitions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Enumerations without Repetitions 1 Talk - Discussion30 Enumerations with Repetitions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Immediate feedback30 Problem solving in Enumerations with Repetitions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 11

Session Outcome: Students will be able to understand different methods of permutations and combinations with constrained repetitions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Enumerations with Repetitions 1 Talk - Discussion30 Enumerations with constrained Repetitions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper30 Problem solving in Enumerations with constrained Repetitions 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 12Session Outcome: Students will be able to understand Binomial Coefficient, Binomial theorem and Multinomial Theorem.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Permutations and Combinations 1 Talk - Discussion30 Binomial Coefficient 1 Marker and Talk - Listen10 Active Learning Program: 2 Fish Bowl30 Binomial theorem and Multinomial Theorem 1 Marker and Talk - Listen10 Active Learning Program: 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 13Session Outcome: Students will be able to understand Principle of Inclusion and Exclusion.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Binomial Coefficient 1 Talk - Discussion50 Problem solving using the Principle of Inclusion and Exclusion 2 Marker and Talk - Listen10 Active Learning Program: 2 Statement – Opinion - Summary10 Pigeonhole Principle 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 1 Video05 Conclusion & Summary Discussion

Session Number: 14Session Outcome: Students will be able to understand the properties of various relations and work with various relations.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Course Outcome -2 1 Talk - Discussion30 Relations, Properties of binary relations 1 Marker and Talk - Listen10 Active Learning Program: 2 Quiz30 Equivalence relations and Ordering relations 1 Marker and Talk - Listen10 Active Learning Program: 2 Immediate Feedback05 Conclusion & Summary Discussion

Session Number: 15

Session Outcome: Students will be able to understand Partial order relation, Lattice and its properties and types.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on relations 1 Talk - Discussion30 Operations on relations, Partial order, Lattice 1 Marker and Talk - Listen10 Active Learning Program: 1 One Minute Paper30 Lattices – types, steps to check the type of Lattice 1 Marker and Talk - Listen10 Active Learning Program: 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 16Session Outcome: Students will be able to solve problems in Lattices.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Lattices 1 Talk - Discussion30 Problems on Lattices 2 Marker and Talk - Listen10 Active Learning Program: 2 Statement – Opinion - Summary30 Problems on Lattices 1 Marker and Talk - Listen10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 17Session Outcome: Students will be able to understand different methods for conversation in Graphs.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Lattices 1 Talk - Discussion30 Graphs – Digraphs, Adjacency matrices, Isomorphism 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper30 Problems on Isomorphism 2 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 18Session Outcome: Students will be able to learn the different methods of Sub-graphs and the structured planar graph notation.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Graphs 1 Talk - Discussion30 Planar graphs, multi graphs, Euler’s formula, Euler’s circuit 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Video30 Searching & Sorting Techniques 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 19Session Outcome: Students will be able to analyze different methods of Graph Theory and their applications.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on types of graphs 1 Talk - Discussion30 Hamiltonian graphs 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 One Minute Paper30 Chromatic Numbers, The Four-Color Problem 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Fish Bowl05 Conclusion & Summary Discussion

Session Number: 20Session Outcome: Students will be able to understand what are trees and its properties, types of trees and their applications.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Course Outcome 3 1 Talk - Discussion30 Trees – Properties, Directed Trees, Rooted trees, Spanning Trees 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Immediate Feedback30 Spanning Trees – DFS, BFS 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Fish Bowl05 Conclusion & Summary Discussion

Session Number: 21Session Outcome: Students will be able to understand Prim’s and Kruskal’s algorithms.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on trees 1 Talk - Discussion30 Spanning Trees – Prim’s algorithm 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching30 Kruskal’s algorithm 1 Marker and Talk, PPT’s - Observe10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 22Session Outcome: Students will be able to generate functions for sequences and expressions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Spanning Trees 1 Talk - Discussion30 Introduction to Generating functions 1 Marker and Talk - Listen10 Active Learning Program: 2 One Minute Paper30 Generating functions – problems 1 Marker and Talk - Listen10 Active Learning Program: 2 One Minute Paper05 Conclusion & Summary Discussion

Session Number: 23Session Outcome: Students will be able to solve Recurrence Relation problems using Substitution method.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on Generating functions 1 Talk - Discussion30 formal power series formulae, Calculating coefficients 1 Marker and Talk - Listen10 Active Learning Program: 2 Immediate Feedback30 Solving Recurrence Relation problems using Substitution method 1 Marker and Talk - Listen10 Active Learning Program: 2 Just in time Teaching05 Conclusion & Summary Discussion

Session Number: 24Session Outcome: Students will be able to solve Recurrence Relation problems using Generating Functions.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on substitution method 1 Talk - Discussion30 Steps for solving Recurrence Relations using Generating Functions 1 Marker and Talk - Listen10 Active Learning Program: 2 Immediate Feedback30 Solving Recurrence Relation problems using Generating Functions 2 Marker and Talk - Listen10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

Session Number: 25Session Outcome: Students will be able to solve Recurrence Relation problems using Characteristic roots.

Time(min) Topic BTL Teaching – Learning Method

05 Attendance10 Recap on generating functions method 1 Talk - Discussion30 Steps for solving Recurrence Relations using Characteristic roots 1 Marker and Talk - Listen10 Active Learning Program: 1 Immediate Feedback30 Solving Recurrence Relation problems using Characteristic roots 2 Marker and Talk - Listen10 Active Learning Program: 2 Just in time teaching05 Conclusion & Summary Discussion

EVALUATION PLAN:

Evaluation Component

Weightage/Marks Date

Duration (Hours) CO 1 CO 2 CO 3 CO 4

COINumber 1 2 3 1 2 3 1 2 3 1 2 3

BTL 1 1 1 2 2 2 2 2 2 2 2 2

Test 1Weightage

(10 %) 27.01.2016 1:30 4 3 3

Max Marks (30) 12 9 9

Test 2Weightage

(10 %) 02.03.2016 1:30 3 3 4

Max Marks (30) 9 9 12

Test 3Weightage

(10 %) 29.03.2016 1:30 3 3 4

Max Marks (30) 9 9 12

Active Learning /

LTCWeightage

(5%) - 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.4 0.4 0.4Max Marks

(25) 2 2 2 2 2 2 2 2 3 2 2 2

Attendance Weightage (5%) - - - - - - - - - - - - -

Semester End Exam

Weightage (60%)

11.05.2016 3:00

4 4 4 4 4 4 4 4 4 8 8 8Max

Marks(60) 4 4 4 4 4 4 4 4 4 8 8 8

Question Number

1(a)or

2(a)

1(b)or

2(b)

1(c)or

2(c)

3(a) or

4(a)

3(b)or

4(b)

3(c) or

4(c)

5(a)or

6(b)

5(b)or

6(b)

5(c) or

6(c)

7(a) 9(a)or

8(a) 10(a)

7(b) 9(b)or

8(b)10(b)

7(c) 9(c) or

8(c) 10(c)

Course Team members, Chamber Consultation Hours and Chamber Venue details:

S.No. Name of Faculty Chamber Consultation Day(s)

Chamber Consultation Timings

for each day

Chamber Consultation

Room No:

Signature of Course faculty

1 Dr C.M.Sheela Rani All working days 12:40 to 1:30 PM C404

2 Dr.K.Subramanyam All working days 4:00 to 5:00 PM C116

3 Dr.S.Satyanarayana All working days 12:40 to 1:30 PM E008

4 Mr.V.Naga Bhushanam All working days 12:40 to 1:30 PM E008

5 Ms.T.Siva Priya All working days 12:40 to 1:30 PM E008

6 Mrs.T.Sajana All working days 12:40 to 1:30 PM C124 (Forensic Lab) – I Floor

7 Mrs. J.Yamini Devi All working days 12:40 to 1:30 PM C504

8 Ms. Yamini Gupta All working days 12:40 to 1:30 PM C404

9 Mr. U Narayankar All working days 12:40 to 1:30 PM C304

Signature of COURSE COORDINATOR:

Recommended by HEAD OF DEPARTMENT:

Approved By: DEAN-ACADEMICS

(Sign with Office Seal)