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DISCRETE MATHEMATICS
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AGI, Dept. of. CSE
ANURAG GROUP OF INSTITUTIONS
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
II Yr CSE –I SEM (2013-2014)
DISCRETE MATHEMATICS
Time Table:
9:00-9:50
MON
DM(B)
TUE
DM(C)
9:50-10:40
11:30-12:20
1:10-2:00
2:00-2-50
DM(D)
DM(C)
WED
DM(A)
DM(D)
THR
FRI
SAT
10:4011:30
DM(B)
DM(B)
DM(D)
DM(B)
DM(D)
DM(A)
DM(C)
DM(C)
DM(B)
DM(D)
DM(A)
2:50-3:40
DM(B)
DM(A)
DM(A)
DM(C)
DM(A)
DM(D)
DM(C)
Faculty:
1. CSE-A
Mr. B.V.Reddy
2. CSE-B
Mr. Dasu Vaman Ravi Prasad
3. CSE-C
Mr. C. Pavan Kumar
4. CSE-D
Mrs. S. Kalyani
Required Text Books:

Discrete Mathematics for Computer Scientists & Mathematicians – 2nd Edition, Joe
L.Mott. Abraham Kandel Theodore P.Baker
REFERENCES
1. Elements of DISCRETE MATHEMATICS – A computer Oriented Approach – CL Liu,
DP Mohapatra. Third Edition, Tata McGraw Hill.
2. Discrete and Combinational Mathematics – An applied introduction – 5th edn. – Ralph.
P.Grimaldi. Pearson Education
AGI, Dept. of. CSE
Course Objectives:

Topics in discrete mathematics have an increasing relevance both of intrinsic interest to
pure mathematicians and in applications such as computer science, networking and
cryptography.
 This course will provide a foundation of various areas of discrete mathematics,
emphasizing their importance and relevance, and encouraging independent and creative
thought in applying the concepts.
Course Outcomes:






Some fundamental mathematical concepts and terminology
How to use and analyze recursive definitions;
How to count some different types of discrete structures;
Techniques for constructing mathematical proofs, illustrated by discrete mathematics
examples.
The student will know and will be able to use mathematical induction to establish
recurrence relations.
The student will know basic concepts, algorithms and problems on graphs and trees and
how to use these structures to organize and process data.
Evaluation Methodology:
S.no
1.
2.
3.
4.
5.
Method of Evaluation
Internal Exam -I
Internal Exam -II
Assignment -I
Assignment -II
External Exam
Examination Dates
10/09/2013 – 12/09/2013
15/11/2013 – 18/11/2013
07/09/2013
13/11/2013
26/11/2013 – 06/12/2013
Marks
20
20
5
5
75
AGI, Dept. of. CSE
UNIT-I
Syllabus:
Foundation - Basics, Sets and Operations of Sets, Relations and Functions, Some Methods of
Proof 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.
Objectives:


Introduce Sets, Relations, Mathematical Logic, especially First Order Logic to students
intending to graduate in Computer Science.
Introduce proof techniques such as Mathematical Induction and Contradiction.
These techniques will come in handy for courses such as Analysis of Algorithms
and Automata Theory.
Micro plan:
S.No
1
2
3
4
5
6
7
8
Topic
No. of
lectures
Basics, Sets and Operations of Sets
1
Relations and Functions
1
Some Methods of Proof and Problem Solving Strategies
2
Fundamentals of Logic, Logical Inferences
2
Methods of Proof of an Implication
1
First Order Logic and Other Methods of Proof
2
Rules of Inference for Quantified Propositions
2
Mathematical Induction
1
TOTAL
12
Assignment:
1. Give truth tables for the following expressions:
a. (s v t) ∧ (￢s v t) ∧ (s v￢t)
b. (s -> t) ∧ (t -> u)
c. (s v t v u) ∧ (s v￢t v u)
AGI, Dept. of. CSE
2. Find at least two more examples of the use of some word or phrase equivalent to “implies”.
3. Find at least two more examples of the use of the phrase “if and only if”.
4. Show that the statements s -> t and ￢s v t are equivalent.
5. Prove the DeMorgan law which states ￢ (p ∧ q) = ￢pv￢q.
6. Show that p ⊕ q is equivalent to (p∧￢q) v (￢p ∧ q).
7. Give a simplified form of each of the following expressions (using T to stand for a statement
that is always true and F to stand for a statement that is always false):
• s v s,
• s ∧ s,
• sv￢s,
• s∧￢s.
8. Use a truth table to show that (s v t) ∧ (u v v) is equivalent to (s∧ u) v (s∧ v) v (t∧ u) v(t∧ v).
What algebraic rule is this similar to?10.A={1,3,5} ,B={2,3} C={4,6} find A X B ,B X A ,(A U B) X
C,(A X B) U(B XC).
11.Draw the Hasse diagram for R={(1,1),(1,2),(1,3)(2,3)(2,4)(3,4)}
12. Explain the properties of relations with examples.
AGI, Dept. of. CSE
UNIT-II
Syllabus:
Elementary Combinations - Basics of Counting, Combination and Permutations, Enumeration
of Combinations and Permutations, Enumerating Combinations and Permutations with
Repetitions, Enumerating Permutations with Constrained Repetitions, Binomial and Multinomial
Theorems, The principle of Inclusion-Exclusion.
OBJECTIVES:
1. Calculate numbers of possible outcomes of elementary combinatorial processes such as
permutations and combinations.
2. Calculate probabilities and discrete distributions for simple combinatorial processes;
calculate expectations.
Micro-Plan:
S.No
1
2
3
4
5
6
7
Topic
No. of
lectures
Basics of Counting
1
Combination and Permutations
1
Enumeration of Combinations and Permutations
2
Enumerating Combinations and Permutations with
Repetitions
2
Enumerating Permutations with Constrained Repetitions
2
Binomial and Multinomial Theorems
2
The principle of Inclusion-Exclusion.
2
Total
12
Assignment:
1. Q 1: Find the number of nonnegative integer solutions of the inequality
x1+x2+x3+…….+x6<10
2. In how many ways can we distribute 7 apples and 6 oranges among 4 children so that
each child gets at least 1 apple?
AGI, Dept. of. CSE
3. Show that C(n-1 +r,r) represents the number of binary numbers that contains (n-1) 1’s
and r 0’s.
4. Find the number of ways of placing 20 identical balls into 5 boxes with atleast one ball
put into each box.
AGI, Dept. of. CSE
UNIT –III
Syllabus:
Recurrence Relations – Generating Functions of Sequences, Calculating Coefficients of
Generating Functions, Recurrence Relations, Solving Recurrence Relations by Substitution and
Generating Functions, the Method of Characteristic Roots, Solutions of Inhomogeneous
Recurrence Relations.
Relations and Digraphs – Relations and Directed Graphs, Special Properties of Binary
Relations, Equivalence Relations, Ordering Relations, Lattice, and Enumerations, Operations and
Relations, Paths and Closures.
Objectives:


Solve problems involving recurrence relations and generating functions.
Interpret, identify, and apply the mathematics associated with graphs, trees,
Weighted graphs, and digraphs.
Micro Plan:
S.No
1
Topic
No. of lectures
Generating Functions of Sequences
1
2
Calculating Coefficients of Generating Functions
2
3
Recurrence Relations, Solving Recurrence Relations
by Substitution and Generating Functions
2
4
The Method of Characteristic Roots
1
5
Solutions of Inhomogeneous Recurrence Relations.
1
6
Relations and Directed Graphs
1
7
Special Properties of Binary Relations
1
8
Equivalence Relations, Ordering Relations
1
9
Lattice, and Enumerations
1
10
Operations and Relations, Paths and Closures.
1
Total
12
AGI, Dept. of. CSE
Assignment:
1. Find a recurrence relation and the initial condition for the sequence 2,10,50,250………
2. A bank pays a certain % of annual interest on deposits, compounding the interest once in 3
months. If a deposit doubles in 6 yrs and 6 months ,what is the annual % of interest paid by the
bank.
3. Find a recurrence relation for the number of binary sequences of length n>=1 that have no
Consecutive 0’s.
4. Find the general solution of the recurrence relation
an-7an-2+10an-4=0,n>=4
AGI, Dept. of. CSE
UNIT-IV
Syllabus:
Graphs - Basic Concepts, Isomorphisms 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.
Objectives:

Apply graph theory models of data structures and state machines to solve problems of
connectivity and constraint satisfaction, for example, scheduling.
Micro Plan:
S.No
1
Topic
No. of lectures
Basic Concepts
1
2
Isomorphism and Sub-graphs
2
3
Trees and Their Properties
1
4
Spanning Trees, Directed Trees, Binary Trees
2
5
Planar Graphs, Euler’s Formula, Multi-graphs and Euler
Circuits
3
6
Hamiltonian Graphs, Chromatic Numbers, The FourColor Problem.
3
Total
12
Assignment:
1. Show that the complete bipartite graph K33 is non planar.
2. A necessary and sufficient condition for a graph G to be a planar is that G does not contain
K5or K33 as a sub graph.
3. Prove that the Petersen graph is non planar.
4. Show that every graph with 4 or fewer vertices is planar.
5. Find the maximum number of edges possible in simple connected planar graph with 4 vertices.
6. What are the applications of graph theory?
7. Explain Chromatic number and Map Coloring Problem with examples.
8. Give the difference between DFS and BFS by considering examples.
9. Prove that a connected graph G has an Euler circuit if and only if all vertices of G are of even
degree.
10. Is there a graph with odd number of vertices and even number of edges that contains an Euler
circuit?
AGI, Dept. of. CSE
UNIT-V
Syllabus:
Boolean Algebras - Introduction, Boolean Algebras, Boolean Functions, Switching
Mechanisms, Minimization of Boolean Functions, Applications of Digital Computer Design.
Network Flows – Graphs as Models of Flow of Commodities, Flows, Maximal Flows and
Minimal Cuts, The Max Flow-Min Cut Theorem, Applications of Matching and Hall’s Marriage
Theorem.
Objectives:
1. An understanding of Boolean algebra sufficient to design combinational and sequential
logic circuits.
2. To understand network flow algorithms and applications
Micro Plan:
S.No
1
Topic
No. of lectures
Introduction, Boolean Algebras, Boolean Functions
2
2
Switching Mechanisms, Minimization of Boolean
Functions,
2
3
Applications of Digital Computer Design
1
4
Graphs as Models of Flow of Commodities
2
5
Flows, Maximal Flows and Minimal Cuts
2
6
The Max Flow-Min Cut Theorem
2
7
Case Study on Network Flow
1
Total
12
Assignment:
1.Simplify the following Boolean expressions to minimum no. of literals.
i. x’y’ + xy + x’y
ii. xy’ + y’z’ + x’z’
iii. x’ + xy + xz’ + xy’z’
iv. (x + y)(x + y’).
2. Obtain the complement of the following Boolean expressions.
i. AB + A(B + C) + B’(B + D)
ii. A + B + A’B’C
AGI, Dept. of. CSE
iii. A’B + A’BC’ + A’BCD + A’BC’D’E
iv. ABEF + ABE’F’ + A’B’EF.
3. a) Express the following functions in sum of minterms and product of maxterms.
i. F (A,B,C,D) = B’D + A’D + BD
ii. F(x,y,z) = (xy + z)(xz + y).
(b) Obtain the complement of the following Boolean expressions.
i. (AB’ + AC’)(BC + BC’)(ABC)
ii. AB’C + A’BC + ABC
iii. (ABC)’(A + B + C)’
iv. A + B’C (A + B + C’).
4. a) Reduce the following Boolean Expressions.
i. AB + A(B + C) + B’(B + D) ii. A +B + A’B’C
iii. A’B + A’BC’ + A’BCD + A’BC’D’E
iv. ABEF + AB(EF)’ + (AB)’EF.
(b) Obtain the Dual of the following Boolean expressions.
i. x’yz’+ x’yz’ + xy’z’ + xy’z
ii. x’yz + xy’z’ + xyz + xyz’
iii. x’z + x’y + xy’z + yz
iv. x’y’z’ + x’yz’ + xy’z’ + xy’z + xyz’.
5.For the below figure find Max Flow-Min Cut Theorem
6. Define Flow. Explain various Models of Flow of Commodities?
AGI, Dept. of. CSE