b.c.a 2017 - pes degree | welcome to pes degreedegree.pes.edu/pdf/2017/bca/discrete mathematics...

B.C.A 2017

PES DEGREE COLLEGE BANGALORE SOUTH CAMPUS Affiliated to Bangalore University

DISCRETE MATHEMATICS BCA105T

MODULE SPECIFICATION SHEET

Course Outline In order to be able to formulate what a computer system is suppose to do, or to prove that it does meet its specification, or to reason about its efficiency, one needs the precision of mathematical notation and techniques. For instance, to specify computational problems precisely one needs to abstract the detail and then mathematical objects such as sets, functions, relations, orders and sequences. To prove that a proposed solution does work as specified, one needs to apply the principle of mathematical logic, and to use proof techniques such as induction. And to reason about the efficiency of an algorithm, one often needs to count the size of complex mathematical objects. The Discrete Mathematics course aims to provide this mathematical background.

<<Course Outline>>

Faculty Details KRISHNAMURTHY H Assistant Professor

Department of Mathematics [email protected]

PHOTO

of 18

1. GENERAL INFORMATION Academic Year : 2017 Semester : I

Title Code Duration

DISCRETE MATHEMATICS BCA105T

Lectures 65

Seminars

Total: 65

Credits

2. PRE REQUIREMENT STATEMENT To be successful in this course student should have mastery of college algebra and be confident in elementary mathematics ‐ be able to develop algorithms and They must also possess problem solving skills.

3. COURSE RELEVANCE This course deals with discrete, or finite, processes and sets of elements. Accordingly, many of the ideas included have direct application of computers. The topics to be discussed are propositional, mathematical logic, sets, functions, relation, matrices, determinants, logarithms, permutation and combination, vectors and analytical geometry of two dimensions.

4. LEARNING OUTCOMES • Some fundamental mathematical concepts and terminology;

• How to use and analyse recursive definitions;

• How to count some different types discrete structure;

• Techniques for constructing mathematical proofs, illustrated by discrete mathematics example.

5. VENUE AND HOURS/WEEK All lectures will normally be held on VIII Floor. Lecture Sessions / Week : 5

of 18

6. MODULE MAP

Class #

CHAPTERS

Topic Details

Cumulative % of

Portions Covered

1

UNIT-II MATRICES AND DETERMINANTS

Introduction to Matrices and Determinants

Author: G.K.Ranganath.

Page No: 1.1-1.3(91-93)

20%

2

Determinants and Adjoint of Matrix


Page No: 1.3-1.4(93-94)

3

Inverse of a Matrix


Page No: 1.3-1.4(93-94)

4

Solution of Linear Equations-Cramer’s Rule


Page No: 1.5(94-97)

5

Solution of System of Linear Equations by Matrix Method


Page No: 1.6(97-100)

6

Solution of System of Linear Equations by Matrix Method


Page No: 1.6(97-100)

7

Eigen Values and Eigen Vectors


Page No: 1.7(100-103)

8

Eigen Values and Eigen Vectors


Page No: 1.7(100-103)

9

Matrix Polynomial


Page No: 1.8(107-109)

10

Verification of Cayley Hamilton Theorem (only 2x2

matrices)


Page No: 1.8(109-113)

of 18

11

Revision on Matrices and Determinants


Page No: 1.1-1.6(91-100)

12

Revision on Eigen value and Eigen Vector, Cayley

Hamilton Theorem


Page No: 1.7-1.8(100-113)

13

UNIT-III LOGARITHMS, PERMUTATION AND COMBINATIONS

Introduction to Logarithms, Definition


Page No: 1.1-1.2(117-119)

40%

14

Properties of Logarithms


Page No: 1.3(119-126)

15

Common Logarithm


Page No: 1.4-1.5(126-132)

16

Common Logarithm


Page No: 1.4-1.5(126-132)

17

Introduction to Permutations and Factorial


Page No: 2.1-2.2(132--135)

18

Fundamental Principles of Counting (Rule of Sum and

Product)


Page No: 2.3(136--140)

19

Fundamental Principles of Counting (Rule of Sum and

Product)


Page No: 2.3(136--140)

20

Permutations


Page No: 2.4(140--150)

of 18

21

Permutations


Page No: 2.4(140--150)

22

Permutations of like Things


Page No: 2.5(150--156)

23

Introduction to Combinations


Page No: 3.1-3.2(157—158)

24

Results related to n𝑪𝒓 (i.e., to C (n, r))


Page No: 3.3(158—171)

25

Results related to n𝑪𝒓 (i.e., to C (n, r))


Page No: 3.3(158—171)

26

UNIT-IV GROUPS AND VECTORS

Introduction to Groups and Binary Operations

(Compositions)


Page No: 1.1-1.2(175—176)

60%

27

Laws of Binary Operation and Algebraic Structures


Page No: 1.3-1.4(176—178)

28

Group and Properties of Group


Page No: 1.5-1.6(178—179)

29

Composition Table for Binary Operation on Finite Set


Page No: 1.7(179—186)

30

Composition Table for Binary Operation on Finite Set


Page No: 1.7(179—186)

31

Modular Systems


Page No: 1.8(187—191)

of 18

32

Subgroup


Page No: 1.9(191—193)

33

Introduction to Scalars and Vectors and Types of Vectors


Page No: 2.1-2.3(194—196)

34

Scalar Multiple of a Vectors and Addition of Two Vectors


Page No: 2.4-2.6(196—201)

35

Plane vectors and Position Vector of a Point in a plane

Space Vectors and the Position Vector of a Point in Space


Page No: 2.7-2.8(201—208)

36

Product of Two Vectors, Scalar Product of Two Vectors

and Geometrical Meaning of Scalar Product


Page No: 2.9-2.11(208—210)

37

Scalar Product in Terms of the Components and Angle

between Two Vectors


Page No: 2.12-2.13(210—216)

38

Vector Product (Cross Product)


Page No: 2.14(217—227)

39

Scalar Triple Product and its Geometrical significance


Page No: 2.15-2.16(228—232)

40

Vector Triple Product


Page No: 2.17(232—237)

41

UNIT-V ANALYTICAL GEOMETRYOF TWO DIMENSIONS

Introduction, Rectangular Cartesian Coordinate System


Page No: 1.1(241—243)

of 18

42

Distance between two points


Page No: 1.3(243-249)

80%

43

Section Formula


Page No: 1.4(251-263)

44

Section Formula


Page No: 1.4(251-263)

45

Area of a Triangle


Page No: 1.5(263-267)

46

Introduction to Locus of a Point


Page No: 2.1-2.2(265-272)

47

Introduction to Straight line and Slope (Gradient ) of line,

Slopes of Parallel Lines and Perpendicular lines and line

joining points


Page No: 3.1-3.4(273—278)

48

Standard Forms of Equation of a straight line


Page No: 3.5(278—292)

49

Standard Forms of Equation of a straight line


Page No: 3.5(278—292)

50

Equation of a line in general form


Page No: 3.6(293—303)

51

Intersection of two lines and angle between the lines


Page No: 3.7-3.8(303—315)

of 18

52

Lines through the intersection of two lines and Length of

the perpendicular from a point to a line


Page No: 3.8-3.11(315—327)

53

UNIT-1 SETS,

RELATIONS AND

FUNCTIONS

Introduction to sets, type and their representations


Page No: 1.1-1.4(3-5)

100%

54

Sub sets, Power set and Universal set.


Page No: 1.4-1.5(5-10)

55

Venn Diagrams and Operations on Sets,


Page No: 1.6(10-14)

56

Some Basis Set Identities and problems on union and

intersection on Sets


Page No: 1.7-1.8(14-29)

57

Ordered Pairs and Cartesian product


Page No: 1.9-(29-37)

58

Relations, Domain, Range and Types of Relations


Page No: 1.10-1.12(37-41)

59

Matrix Representation and Inverse Relation


Page No: 1.13-1.14(45-49)

60

Functions and Types of Functions


Page No: 1.15-1.16(50-51)

61

Introduction to Mathematical Logic, Propositions


Page No: 2.1-2.3(60-61)

62

Fundamentals of Forming the Compound Propositions


Page No: 2.4(61-63)

of 18

63

Logical Equivalence


Page No: 2.5(61-63)

64

Converse, Inverse and Contra positive of a Conditional


Page No: 2.6(72-77)

65

Tautology and Contradiction, Switching Circuits, Basic

Series Network and Basic Parallel Network


Page No: 2.7-29(79-81)

7. RECOMMENDED BOOKS/JOURNALS/WEBSITES

A. PRESCRIBED TEXTBOOK

a. Grewal,B.S.Higher engineering Mathematics, 36th Edition

B. REFERENCE BOOKS

a. Satyrs S.S, Engineering Mathematics b. Peter V.O’Neil. Advanced Engineering Mathematics, 5th Edition.

8. ASSIGNMENT(S) ASSIGNMENT 01

1. If 2A+B=[4 4 77 3 4

], A-2B=[−3 2 11 −1 2

] then find A and B.

2. If A=[−1 3−2 4

], find 𝐴−2 using Cayley Hamilton theorem.

3. Solve the system of equation 5x+2y=4, 7x+3y=5 using Matrix method.

4. Solve using Cramer’s rule

3x+y+z=3

2x+2y+5z=-1

X-3y-4z=2

5. Find the Eigen values and Eigen vectors of A=(1 43 2

)

6. If A=(2 14 −2

), B=(4 32 −1

), find AB.

ASSIGNMENT 02

1. If logx-2log 6

7 =

1

2 log

81

16 - log

27

196 find x.

2. Find the number of different signals that can be generated by arranging atleast 3flags in

order (one below the other) on a vertical staff, if 6 different flags are available.

of 18

3. In how many ways can the letters of the word ASSASSINATION be arranged so that all

the S’s are together. 4. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be

done when the committee consist of (i) Exactly 3 girls (ii) atleast 3 girls (iii) atmost 3

girls. 5. In how many ways the letters of the word “EVALUATE” be arranged so that all vowels

are together. ASSIGNMENT 03

1. Prove that G={1,5,7,11} is a group under multiplication modulo 12. 2. If G={3𝑛 ∶ 𝑛𝜖𝑧}, prove that G is abelian group under multiplication. 3. Find the area of the triangle whose vertices are A(1,2,3), B(2,5,1) and C(-1,1,2) using

vector method. 4. If the vectors 2i-3j+mk, 2i+j-k and 6i-j+2k are coplanar. find m 5. Find the equation of the straight line passing through (2,5) and having slope 4.

ASSIGNMENT 04 1. If A={1,2,3,4},B= {3,4,5,6}, C={5,6,7,8} and D={7,8,9,10} find

(a) AUB (b) A∩ 𝐵 (c) AUBUD (d) A-B (e) A∩B∩D

2. If A={2,3,4,8},B={1,3,4} and U={0,1,2,3,4,5,6,7,8,9}.Verify A-B=A∩ �̅�

3. If A={𝑎, 𝑏, 𝑐, 𝑑},B={𝑐, 𝑑},C={𝑑, 𝑒}, find A-B ,(A-B) ∩(B-C),BXC

4. If R->R is defined by f(x) = 2x+5, prove that f is one-one and onto.

5. If R->R is defined by f(x) = 4x+3, prove that f is invertible.

6. Define tautology and contradiction. Examine whether the following compound

proposition is a tautology or a contradiction. p/\ (qvr)<--->(p/\q) V (p/\r)

10. THEORY ASSESSMENT

A. WRITTEN EXAMINATION

The Theory Examination is for 100 Marks which will be held for duration of 3 Hrs.

The Scheme and Blue Print will be released to the students once the Bangalore

University releases it.

B. CONTINUOUS ASSESSMENT The Continuous Assessment is conducted as per the following parameters.

Parameter MARKS WEIGHTAGE % 22 MARKS

Internal Test 50MARKS 75% 16.5 MARKS

of 18

Assignment 10 MARKS 12.5% 2.5 MARKS

Class Test 10 MARKS 12.5% 2.5 MARKS

Total 70 MARKS 100% 22 MARKS

The students are hereby required to note that every internal test weightage will calculated for 22 Marks. This includes timely submission of assignments and attending class tests as conducted. The Sum of Best Two Performances in Internal Terms will be taken.

Parameter MARKS

Internal Test 01 22 MARKS



Final Internal Marks (Sum of Best Two Marks Of The Three Internal Tests) 44 MARKS

Attendance >95 % : 06 Marks 90 - 95 % : 05 Marks 85 - 90 % : 04 Marks 80 - 85 % : 03 Marks 75 - 80 % : 02 Marks

06 MARKS

Total 50 MARKS

11. ASSESSMENT / ASSIGNMENT / CLASS TEST / ACTIVITY PLANNER

Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Internal Test T1 T2 T3

Assignments

Submission A1 A2 A3

Class Test C1 C2 C3

Legend Meaning Test Topics Examinable

T1, T2,T3 Internal Tests T1 Class 1 – 20

A1, A2, A3, A4, A5, A6 Assignments T2 Class 21 – 42

C1,C2,C3 Class Test T3 Class 42 – 65

ME Mock Exam

of 18

12. QUESTION BANK 1. If A={1,2,3,4},B= {3,4,5,6}, C={5,6,7,8} and D={7,8,9,10} find

(b) AUB (b) A∩ 𝐵 (c) AUBUD (d) A-B (e) A∩B∩D

2. If A={2,3,4,8},B={1,3,4} and U={0,1,2,3,4,5,6,7,8,9}.Verify A-B=A∩ �̅�

3. If A={𝑎, 𝑏, 𝑐, 𝑑},B={𝑐, 𝑑},C={𝑑, 𝑒}, find A-B ,(A-B) ∩(B-C),BXC

4. If R->R is defined by f(x) = 2x+5, prove that f is one-one and onto.

5. If R->R is defined by f(x) = 4x+3, prove that f is invertible.

6. Define tautology and contradiction. Examine whether the following compound

proposition is a tautology or a contradiction. p/\ (qvr)<--->(p/\q) V (p/\r)

7. If 2A+B=[4 4 77 3 4

], A-2B=[−3 2 11 −1 2

] then find A and B.

8. If A=[−1 3−2 4

], find 𝐴−2 using Cayley Hamilton theorem.

9. If logx-2log 6

7 =

1

2 log

81

16 - log

27

196 find x.

10. Find the number of different signals that can be generated by arranging atleast

3flags in order (one below the other) on a vertical staff, if 6 different flags are

available. 11. In how many ways can the letters of the word ASSASSINATION be arranged so

that all the S’s are together. 12. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways

can this be done when the committee consist of (i) Exactly 3 girls (ii) at least 3

girls (iii) at most 3 girls. 13. In how many ways the letters of the word “EVALUATE” be arranged so that all

vowels are together.

14. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways

can this be done when the committee consist of (i) Exactly 3 girls (ii) atleast 3

girls (iii) at most 3 girls. 15. In how many ways the letters of the word “EVALUATE” be arranged so that all

vowels are together 16. Prove that G={1,5,7,11} is a group under multiplication modulo 12. 17. If G={3𝑛 ∶ 𝑛𝜖𝑧}, prove that G is abelian group under multiplication. 18. Find the area of the triangle whose vertices are A(1,2,3), B(2,5,1) and C(-1,1,2)

using vector method. 19. If the vectors 2i-3j+mk, 2i+j-k and 6i-j+2k are coplanar. find m 20. Find the equation of the straight line passing through (2,5) and having slope 4

of 18

13. MODEL QUESTION PAPERS

b.c.a 2017 - pes degree | welcome to pes degreedegree.pes.edu/pdf/2017/bca/discrete mathematics...

Documents