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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
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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
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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
Author: G.K.Ranganath.
Page No: 1.3-1.4(93-94)
3
Inverse of a Matrix
Author: G.K.Ranganath.
Page No: 1.3-1.4(93-94)
4
Solution of Linear Equations-Cramer’s Rule
Author: G.K.Ranganath.
Page No: 1.5(94-97)
5
Solution of System of Linear Equations by Matrix Method
Author: G.K.Ranganath.
Page No: 1.6(97-100)
6
Solution of System of Linear Equations by Matrix Method
Author: G.K.Ranganath.
Page No: 1.6(97-100)
7
Eigen Values and Eigen Vectors
Author: G.K.Ranganath.
Page No: 1.7(100-103)
8
Eigen Values and Eigen Vectors
Author: G.K.Ranganath.
Page No: 1.7(100-103)
9
Matrix Polynomial
Author: G.K.Ranganath.
Page No: 1.8(107-109)
10
Verification of Cayley Hamilton Theorem (only 2x2
matrices)
Author: G.K.Ranganath.
Page No: 1.8(109-113)
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11
Revision on Matrices and Determinants
Author: G.K.Ranganath.
Page No: 1.1-1.6(91-100)
12
Revision on Eigen value and Eigen Vector, Cayley
Hamilton Theorem
Author: G.K.Ranganath.
Page No: 1.7-1.8(100-113)
13
UNIT-III LOGARITHMS, PERMUTATION AND COMBINATIONS
Introduction to Logarithms, Definition
Author: G.K.Ranganath.
Page No: 1.1-1.2(117-119)
40%
14
Properties of Logarithms
Author: G.K.Ranganath.
Page No: 1.3(119-126)
15
Common Logarithm
Author: G.K.Ranganath.
Page No: 1.4-1.5(126-132)
16
Common Logarithm
Author: G.K.Ranganath.
Page No: 1.4-1.5(126-132)
17
Introduction to Permutations and Factorial
Author: G.K.Ranganath.
Page No: 2.1-2.2(132--135)
18
Fundamental Principles of Counting (Rule of Sum and
Product)
Author: G.K.Ranganath.
Page No: 2.3(136--140)
19
Fundamental Principles of Counting (Rule of Sum and
Product)
Author: G.K.Ranganath.
Page No: 2.3(136--140)
20
Permutations
Author: G.K.Ranganath.
Page No: 2.4(140--150)
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21
Permutations
Author: G.K.Ranganath.
Page No: 2.4(140--150)
22
Permutations of like Things
Author: G.K.Ranganath.
Page No: 2.5(150--156)
23
Introduction to Combinations
Author: G.K.Ranganath.
Page No: 3.1-3.2(157—158)
24
Results related to n𝑪𝒓 (i.e., to C (n, r))
Author: G.K.Ranganath.
Page No: 3.3(158—171)
25
Results related to n𝑪𝒓 (i.e., to C (n, r))
Author: G.K.Ranganath.
Page No: 3.3(158—171)
26
UNIT-IV GROUPS AND VECTORS
Introduction to Groups and Binary Operations
(Compositions)
Author: G.K.Ranganath.
Page No: 1.1-1.2(175—176)
60%
27
Laws of Binary Operation and Algebraic Structures
Author: G.K.Ranganath.
Page No: 1.3-1.4(176—178)
28
Group and Properties of Group
Author: G.K.Ranganath.
Page No: 1.5-1.6(178—179)
29
Composition Table for Binary Operation on Finite Set
Author: G.K.Ranganath.
Page No: 1.7(179—186)
30
Composition Table for Binary Operation on Finite Set
Author: G.K.Ranganath.
Page No: 1.7(179—186)
31
Modular Systems
Author: G.K.Ranganath.
Page No: 1.8(187—191)
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32
Subgroup
Author: G.K.Ranganath.
Page No: 1.9(191—193)
33
Introduction to Scalars and Vectors and Types of Vectors
Author: G.K.Ranganath.
Page No: 2.1-2.3(194—196)
34
Scalar Multiple of a Vectors and Addition of Two Vectors
Author: G.K.Ranganath.
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
Author: G.K.Ranganath.
Page No: 2.7-2.8(201—208)
36
Product of Two Vectors, Scalar Product of Two Vectors
and Geometrical Meaning of Scalar Product
Author: G.K.Ranganath.
Page No: 2.9-2.11(208—210)
37
Scalar Product in Terms of the Components and Angle
between Two Vectors
Author: G.K.Ranganath.
Page No: 2.12-2.13(210—216)
38
Vector Product (Cross Product)
Author: G.K.Ranganath.
Page No: 2.14(217—227)
39
Scalar Triple Product and its Geometrical significance
Author: G.K.Ranganath.
Page No: 2.15-2.16(228—232)
40
Vector Triple Product
Author: G.K.Ranganath.
Page No: 2.17(232—237)
41
UNIT-V ANALYTICAL GEOMETRYOF TWO DIMENSIONS
Introduction, Rectangular Cartesian Coordinate System
Author: G.K.Ranganath.
Page No: 1.1(241—243)
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42
Distance between two points
Author: G.K.Ranganath.
Page No: 1.3(243-249)
80%
43
Section Formula
Author: G.K.Ranganath.
Page No: 1.4(251-263)
44
Section Formula
Author: G.K.Ranganath.
Page No: 1.4(251-263)
45
Area of a Triangle
Author: G.K.Ranganath.
Page No: 1.5(263-267)
46
Introduction to Locus of a Point
Author: G.K.Ranganath.
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
Author: G.K.Ranganath.
Page No: 3.1-3.4(273—278)
48
Standard Forms of Equation of a straight line
Author: G.K.Ranganath.
Page No: 3.5(278—292)
49
Standard Forms of Equation of a straight line
Author: G.K.Ranganath.
Page No: 3.5(278—292)
50
Equation of a line in general form
Author: G.K.Ranganath.
Page No: 3.6(293—303)
51
Intersection of two lines and angle between the lines
Author: G.K.Ranganath.
Page No: 3.7-3.8(303—315)
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52
Lines through the intersection of two lines and Length of
the perpendicular from a point to a line
Author: G.K.Ranganath.
Page No: 3.8-3.11(315—327)
53
UNIT-1 SETS,
RELATIONS AND
FUNCTIONS
Introduction to sets, type and their representations
Author: G.K.Ranganath.
Page No: 1.1-1.4(3-5)
100%
54
Sub sets, Power set and Universal set.
Author: G.K.Ranganath.
Page No: 1.4-1.5(5-10)
55
Venn Diagrams and Operations on Sets,
Author: G.K.Ranganath.
Page No: 1.6(10-14)
56
Some Basis Set Identities and problems on union and
intersection on Sets
Author: G.K.Ranganath.
Page No: 1.7-1.8(14-29)
57
Ordered Pairs and Cartesian product
Author: G.K.Ranganath.
Page No: 1.9-(29-37)
58
Relations, Domain, Range and Types of Relations
Author: G.K.Ranganath.
Page No: 1.10-1.12(37-41)
59
Matrix Representation and Inverse Relation
Author: G.K.Ranganath.
Page No: 1.13-1.14(45-49)
60
Functions and Types of Functions
Author: G.K.Ranganath.
Page No: 1.15-1.16(50-51)
61
Introduction to Mathematical Logic, Propositions
Author: G.K.Ranganath.
Page No: 2.1-2.3(60-61)
62
Fundamentals of Forming the Compound Propositions
Author: G.K.Ranganath.
Page No: 2.4(61-63)
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63
Logical Equivalence
Author: G.K.Ranganath.
Page No: 2.5(61-63)
64
Converse, Inverse and Contra positive of a Conditional
Author: G.K.Ranganath.
Page No: 2.6(72-77)
65
Tautology and Contradiction, Switching Circuits, Basic
Series Network and Basic Parallel Network
Author: G.K.Ranganath.
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.
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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
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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
Internal Test 02 22 MARKS
Internal Test 03 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
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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
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13. MODEL QUESTION PAPERS
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