Hand out dm

II CSE – II SEM M.NARMADHA 2019-20
Department of CSE DRK COLLEGE OF ENGINEERING AND TECHNOLOGY- 1 -
DRK COLLEGE OF ENGINEERING AND TECHNOLOGY
HYDERBAD
Handout
Academic Year: 2019 – 20 , Semester: II
Faculty: M.NARMADHA Designation: Asst.Professor Dept.: CSE
Class: II B.Tech Branch: CSE Subject: Discrete Mathematics
1. Course objectives
 Introduces the elementary discrete mathematics for computer science and engineering.
 Topics include formal logic notation, methods of proof, induction, sets, relations, graph
theory, permutations and combinations, counting principles; recurrence relations and
generating functions.
2. SuggestedResearchand Survey Themes
Combinatorial Optimization has become more and more important during the last decades, due to
its immense significance for applications. At the Research Institute for Discrete Mathematics, it has
always played a major role.
Chip Design is probably the most interesting and varied range of application of mathematics in
general. Modern highly complex chips cannot be designed without the use of methods of discrete
mathematics.
3. Contents beyond syllabus
 Predicates and quantifiers
 Algorithms, Induction and Recursion
4. Prescribedtext books
1. Discrete Mathematics and its Applications with Combinatorial and Graph Theory- Kenneth H
Rosen, 7th Edition, TMH.
5. Reference books
1. Discrete Mathematical Structures with Applications to Computer Science-J.P. Tremblay and
R. Manohar, TMH,
2. Discrete Mathematics for Computer Scientists & Mathematicians: Joe L. Mott, Abraham
Kandel, Teodore P. Baker, 2nd ed, Pearson Education.
3. Discrete Mathematics- Richard Johnsonbaugh, 7Th Edn., Pearson Education.
4. Discrete Mathematics with Graph Theory- Edgar G. Goodaire, Michael M. Parmenter.

5. Discrete and Combinatorial Mathematics - an applied introduction: Ralph.P. Grimald, 5th
edition, Pearson Education.
6. URLs and other e-learning resources
 https://nptel.ac.in/courses/106106183/#
 https://nptel.ac.in/courses/106106094/
 https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematic
s_propositional_logic.htm
 http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/
7. Brief history and current developments in the subject area
 Discrete mathematics (DM) is a branch of mathematics which deals with mathematical
operations within a subset of the real numbers where data is considered more as objects rather
than numbers. Characteristically DM provides us problem solving solutions for only distinct
and countable quantities which stands clearly in contrast to e.g. the mathematical analysis
which encompasses continuous functions applied on uncountable and infinitive quantities.
The term “discrete” (Latin discretum) highlights exactly this difference. Even though some
sections of DM like e.g. number- or graph theory are relatively old, discrete mathematics has
been overshadowed for centuries by the “continuous” mathematics due to the invention of
infinitesimal calculus and its multifaceted applications within natural sciences (particularly in
physics).
 In 1847, over hundred years later, Leibniz’ idea was finally realized by an english, self-taught
mathematicians named George Boole (1815 - 1855). Boole extended the concept from
Leibniz. He created the first algebra of logic and published his thoughts on this topic in his
works The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of
Thought (1854). He found out that the symbols of logic (today known as the subject of
symbolic logic) behave exactly like those in algebra and he used algebraic symbols to express
4 (Epp, 2010) Page 5 / 9 logical relations. He claimed that only three operations (AND, OR
and NOT) are needed to perform all other logical functions. The terminology Boolean algebra
was later suggested by Sheffer in 19135 . Today George Boole (along with Charles Babbage,
who designed the first mechanical calculator, confer Difference Engine) is considered as one
of the grandfathers of computing.
 Mathematicians have always worked with logic and symbols, but for centuries the underlying
laws of logic were taken for granted, and never expressed symbolically. Mathematical logic,
also known as symbolic logic, was developed when people finally realized that the tools of
mathematics can be used to study the structure of logic itself. Areas of research in this field
have expanded rapidly, and are usually subdivided into several distinct subfields.

Proof theory and constructive mathematics
Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in
mathematics. The most famous result in the field is encapsulated in Gödel's incompleteness
theorems. A closely related and now quite popular concept is the idea of Turing
machines. Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic
itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not"
until they have actually exhibited a circle and measured its roundness.
Model theory
Model theory studies mathematical structures in a general framework. Its main tool is first-order
logic.
Set theory
A set can be thought of as a collection of distinct things united by some common feature. Set theory
is subdivided into three main areas. Naive set theory is the original set theory developed by
mathematicians at the end of the 19th century. Axiomatic set theory is a rigorous axiomatic theory
developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set
theory. It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves
only as motivation for the axioms. Internal set theory is an axiomatic extension of set theory that
supports a logically consistent identification of (enormously large) and infinitesimal(unimaginably
small) elements within the real numbers. See also List of set theory topics.
8. Lecture schedule / lessonplan
Name of the Topic No. of Periods
UNIT - I
1. The Foundations 1
2. Logic and Proofs 1
3. Propositional Logic 2
4. Applications of Propositional Logic 1
5. Propositional Equivalence 1
6. Predicates and Quantifiers, Nested Quantifiers 2
7. Rules of Inference 1
8. Introduction to Proofs 1
9. Proof Methods and Strategy 2
UNIT – II
1. Basic Structures, 1
2. Sets, 1
3. Functions, 2
4. Sequences, 1
5. Sums, 1
6. Matrices and Relations Sets, . 1
7. Functions, 1
8. Sequences & Summations, 1

9. Cardinality of Sets and Matrices Relations, 1
10. Relations and Their Properties, 1
11. n-ary Relations and Their Applications, 1
12. Representing Relations, 1
13. Closures of Relations, 2
14. Equivalence Relations, 1
15. Partial Orderings. 1
UNIT - III
1. Algorithms, Induction and Recursion: 1
2. Algorithms, The Growth of Functions, 1
3. Complexity of Algorithms Induction and Recursion: 2
4. Mathematical Induction, 1
5. Strong Induction and Well-Ordering, 1
6. Recursive Definitions and Structural Induction, 1
7. Recursive Algorithms, 2
8. Program Correctness 2
UNIT - IV
1. Discrete Probability and Advanced Counting Techniques: 2
2. An Introduction to Discrete Probability, 1
3. Probability Theory, 1
4. Baye’s Theorem, 2
5. Expected Value and Variance Advanced Counting Techniques: 1
a. Recurrence Relations, 1
b. Solving Linear Recurrence Relations, 1
c. Divide-and-Conquer Algorithms and Recurrence Relations, 2
d. Generating Functions, 1
e. Inclusion Exclusion 1
f. Applications of Inclusion-Exclusion 1
UNIT – V
a. Graphs: Graphs and Graph Models, 1
b. Graph Terminology and Special Types of Graphs, 1
c. Representing Graphs and Graph Isomorphism, 2
d. Connectivity, 1
e. Euler and Hamilton Paths, 1
f. Shortest-Path Problems, 1
g. Planar Graphs, 1
h. Graph Coloring. 1
i. Trees: 1
j. Introduction to Trees, 1
k. Applications of Trees, 1
l. Tree Traversal, 1
m. Spanning Trees, 1
n. Minimum Spanning Trees 2
TOTAL PERIODS: 70

9 Seminars / group discussions, ifany and their schedule
Graphs and Graph Model
Introduction to Trees
10.Distribution and Weightage ofMarks
Assessment Component Marks Schedule
Final Marks
Assignment Test#1 (AT#1) 5 After end on Unit#1
25
(average of SE1,SE2)Sessional Exam#1 (SE#1) 20
At the end of Unit#1
& 2
Assignment Test#2 (AT#2) 5 After and on Unit#3
Sessional Exam#2 (SE#2) 20
At the end of Unit#3
& 4
Semester End Exam 75 Semester End
11. Attendance Requirements:
A student shall be eligible to appear for University examinations if he acquires a minimum of 75%
of attendance.
12. Academic Calendar
The Proposed Academic Calendar for III Year II Semester B.Tech courses during the
Academic year 2019-20 is detailed below.
B.TECH IV YEAR II Semester
Description From To Weeks
Commencement of Class
Work
16th Dec,2010 -
I Unit of Instructions 16th dec,2010 9th Feb,2010 8W
I Mid Examinations 10th Feb,2020 12th Feb,2020 1W
II Unit of Instruction 13th Feb,2020 7th april,2020 8W
II Mid Examinations 8th April,2020 11th April,2020 1W
Preparation & Practical 13th April,2020 18th April,2020 2W
End Examinations 20th April,2020 2nd May,2020 2W
13. Assignments and tutorials questions
Unit-I
Short Questions
1 Define statement and atomic statement?
2 Explain logical equivalence with an example?

3 Describe the tautology?
4 Apply the converse, inverse and contra positive of the following propositions: P -> (Q -> R)
5 Interpret that Pv[P^(PvQ)] and P is a logically equivalent without using truth table ?
6 Explain P↑Q in terms of “↓” ?
7 Define predicate and predicate logic?
8 Define contradiction and provide a proof by contradiction of the following statements for
every
integer ‘n’ ,if n2 is odd then ‘n’ is odd.
9 Define converse, contra positive and inverse of implication?
10 Analyze and symbolize the following statements:
a) all men are good
b) no men are good
11 Examine the disjunctive normal form of the formula: P↔Q?
12 Describe the value of: P↔Q in terms of {~,v} only ?
13 Explain about the free and bound variables?
14 Illustrate that if ‘m’ is an even integer then m+7 is an odd integer?
15 Demonstrate the truth table for conjunction and conditional statements?
16 Construct the truth table for p->(q->r)?
17 Show that ~(p->q)->p?
18 Construct the statements R: Mark is rich. H:Mark is happy write the following statements
in
symbolic form
a) mark is poor but happy
b)mark is happy but poor
19 Construct the following statement in symbolic form: “the crop will be destroyed if there is a
flood”.
20 Show that R→S can be derived from the premises P→(Q→S), ~R v P and Q
Long Questions
1 a) Explain conditional proposition with a suitable example.
b) Explain logical equivalence with an example.
2 a) Define tautology? Show that [(p->q)->r]->[(p->q)->(p- >r)]is a tautology or not ?
(b) Define the converse, inverse and contra positive of the following propositions:
i. P -> (Q -> R)
ii. (P ^ (P -> Q) ) -> Q.
3 Show that S v R is a tautologically implied by ( p v q ) ^ ( p
With reference to automatic theorem proving.
4 Show that RVS is valid conclusion from the premises:
CVD,(CvD) ~H
5 Show that i)~(P↑Q)↔~P↓~Q ii)~(P↓Q)↔~P↑~Q without using truth table ?
Express p->(~p->q) i)in terms of ‘↑’ only ii)in terms of ‘↓’

6 (a) Describe the proposition (p ^ q) ~ (p v q) is a contradiction.
( b) Symbolize the following statements:
all men are good
no men are good some
men are good
some men are not good
7 (a) Construct the disjunctive normal form of the formula:P↔Q?
(b) Construct the value of: P↔Q in terms of {~,v} only ?
8 Explain about the free and bound variables. With an examples?
9 Show that if ‘m’ is an even integer then m+7 is an odd integer?
ii)write each of the following in symbolic form
a)all monkeys have tails b)no monkey have tail
10 Construct tautology? Show that [(p->q)->r]->[(p->q)->(p->r)] is a tautology or not ?
Unit-II
Short Questions
1. Let us consider the set T of triangles in a plane. Let us define a relation R In T as R={(a, b)/ (a,
bЄT and a is similar to b} We have to show that relation R is an equivalence.
2. Let x = {1, 2, 3, … 7} and R = {(x, y) / x – y is divisible by 3} Show that R is an equivalence
relation
3. ) showthat ‘s’ is a valid conclusion from the given premisesp ~q, q v r, ~sp, ~r.
b) Negate each of the followingstatements.
i) x,yp(x,y).
ii) y,x,z,p(x,y,z).
Long Questions
1. Find the number of integers between 1 and 500 they are not divisible by any of the integers 2,3, 5
and 7. (APR/AMAY2017) Prove that the following result 1 1.2 + 1 2.3 + 1 3.4 + ⋯ + 1 𝑛(𝑛+1) = 𝑛
𝑛+1
2. Show that 1 2 + 2 2 + 3 2 + ⋯ + 𝑛 2 = 𝑛(𝑛+1)(2𝑛+1) 6 , 𝑛 ≥ 1 by mathematical induction.
(APR/MAY2015)
3. Prove that 𝑛 3 + 2𝑛 is divisible by 3 for 𝑛 ≥ 1 .
4. Solve 𝑎𝑛 = 2𝑎𝑛−1 + 5𝑎𝑛−2 − 6𝑎𝑛−3 𝑤𝑖𝑡ℎ 𝑎0 = 7, 𝑎1 = −4 , 𝑎2 = 8. (APR/MAY 2015) 5. Use
generating functions to solve the recurrence relation 𝑎𝑛 = 3𝑎𝑛−1 + 2 with 𝑎0 = 1 (NOV/DEC 2016)

Unit-III
Short Questions
1. Explain about any 4 properties of a relations
2. Let a={1,2,3,4} and r={(1,1),(1,2),(2,1),(2,2),(3,1),(3,3),(1,3),(4,1),(4,4)} be a relation on
a. R. Is r an equivalence relation?
3. In how many ways can four students be selected out of twelve students.
a. If two particular students are not included at all?
b. Two particular students included?
4. How many four digit numbers can be formed using the digits 0,1,2,3,4,5 if
5. Repetition of digit is not allowed.
6. Repetition of digits is allowed.
7. . Find the number of non negative integral solutions to n1+n2+n3+n4=20.
8. Find the term independent of x in the expansion of (x² + 1/x)12.
9. Find the coefficient of a6 b3c3 in the expansion of (a+b+2c)12.
10. Solve the recurrence relation using substitution method
a. An = an-1 + 5n ,a0 = 1.
11. Solve the recurrence relation using generating functions.
a. An - 6an-1 + 12a n-2 – 8an-3 = 0 .
12. Find the size of an r–regular (p,q) graph?
13. Explain BFS and DFS with an example?
Long Questions
1. A simple graph with n vertices and k components can have atmost (𝑛−𝑘)(𝑛−𝑘+1) 2
2. State and Prove Hand Shaking Theorem.
3. Prove that a simple graph with n vertices must be connected if it has more than
(𝑛−1)(𝑛−2) 2
4. Give an example of a graph which is (a) Eulerian but not Hamiltonian (b) Hamiltonian
but not Eulerian
5. Draw the graphs for the following
i. Both Eulerain and Hamiltonian.
ii. non Euleraian and non Hamiltonian.
6. Define Adjacency matrix with an example.
7. If G is a simple connected graph n vertices with n≥ 3 , such that every degree of vertex in
G is at least 𝑛 2 , then prove that G is an Hamiltonian cycle.

Unit – IV
Short Questions
1.Define isomorphism.
2. If (G, ∗) is abelian for any a, b ∈ 𝑮, (a∗b)2 = a2 ∗ b 2 .
3. State Lagranges theorem.
4. Give an example of semi group which is not a monoid.
5. If (G, ∗) is abelian for any a, b ∈ 𝑮, (a∗b)2 = a2 ∗ b 2 .
6. State the necessary and sufficient condition that a non- empty subset H of a group G to be
a subgroup. www.rejinpaul.com
7. Prove that intersection of two subgroups of a group is also a subgroup of the group.
8. Define group.
9. Define subgroup.
10. Define homomorphism.
11. Define semigroups and monoid.
12. Define normal subgroup.
13. Define rings.
14. Define fields.
15. Prove that identity element of a group is unique.
Long Questions
1. Prove that the intersection of two normal subgroup of G is again a normal subgroup of G
2. State and Prove Lagrange’s theorem on groups.
3. S.T. intersection of any two congruence relation on a set 𝐴 is again an congruence relation
on 𝐴.
4. The necessary and sufficient condition that a non-empty subset 𝐻 of a group 𝐺 be a
subgroup is 𝑎 ∈ 𝐻, 𝑏 ∈ 𝐻 ⇒ 𝑎 ∗ 𝑏 −1 ∈ 𝐻 .
5. Let G be a group and 𝑎 ∈ 𝐺 . Let 𝑓: 𝐺 → 𝐺 be given by 𝑓(𝑥) = 𝑎 𝑥 𝑎 −1 for all 𝑥 ∈ 𝐺 .
Prove that 𝑓 is an isomorphism of G onto G.
6. If H and K are subgroup of G, then prove that 𝐻 ∪ 𝐾 is a subgroup of G if and only if
either 𝐻 ⊆ 𝐾 or 𝐾 ⊆ 𝐻
7. State and Prove Cayley’s theorem
8. State and Prove Lagrange’s theorem.
9. Let (𝐺,∗) and (𝐻,△) be groups and 𝑔: 𝐺 → 𝐻 be a homomorphism.
Then prove that the kernel of g is a normal sub-group.
10. State and prove fundamental theorem on homorphismof groups.
11. Prove that the intersection of two normal subgroups is a normal subgroup.
12. If 𝑎 and 𝑏 are any two elements of a group (𝐺,∗) ,then S.T. 𝐺 is an Abelian group if and
only if (𝑎 ∗ 𝑏) 2 = 𝑎 2 ∗ 𝑏 2

Unit-V
Short Questions
1.Define Boolean algebra.
2. Define lattice.
3. Write any two properties of lattices.
4. Define poset.
5. Define sublattices.
6. Prove that every distributive lattice is modular.
7. Prove that in a distributive lattice ,complement of an element is unique.
8. Prove that in a Boolean algebra, a=b iff ab1+a1b=0.
9. Show that in a distributive lattice, if complement of an element exists then it must be
unique..
10. Draw the Hasse diagram for the relation divisibility of (S, /) where S= {1,2,3,4,6,8,12}.
11. Tabulate the properties of Boolean algebra.
12. Define a partially ordered set and draw the Hasse Diagram for (P(A),≤), Where
A={a,b,c}
13. Draw the Hasse diagram for the set of partitions of 5. 14. Define lattice homomorphism.
15. Define Sub-Boolean algebra
Long Questions
1. Show that the operation of meet are join on a lattice are associative.
2. Draw Hasse diagram of all lattices with up to five elements.
3. Prove that every chain is a distributive lattice.
4. Show that in a distributive and complemented lattice 𝑎 ≤ 𝑏 ⟺ 𝑎 ∗ 𝑏 ′ = 0 ⟺ 𝑎 ′ ⊕ 𝑏 = 1
⟺ 𝑏 ′ ≤ 𝑎
5. If (𝐿,∧,∨) is a complemented distributive lattice, the the De Morgan’s laws are valid.
6. Show that in a lattice if 𝑎 ≤ 𝑏 ≤ 𝑐 ,then (1) 𝑎 ⊕ 𝑏 = 𝑏 ∗ 𝑐 (2) (𝑎 ∗ 𝑏) ⊕ (𝑏 ∗ 𝑐) = 𝑏 = (𝑎 ⊕
𝑏) ∗ (𝑐)
7. In any Boolean algebra, show that (𝑎 + 𝑏 1 )(𝑏 + 𝑐 1 )(𝑐 + 𝑎 1 ) = (𝑎 1 + 𝑏)(𝑏 1 + 𝑐)(𝑐 1 +
𝑎)
8. State and Prove De Morgan’s law in a complemented distributive lattice.
9. Show that in a lattice if 𝑎 ≤ 𝑏 and 𝑐 ≤ 𝑑, then 𝑎 ∗ 𝑐 ≤ 𝑏 ∗ 𝑑 and 𝑎 ⊕ 𝑐 ≤ 𝑏 ⊕ 𝑑.
10. In a distributive lattice prove that 𝑎 ∗ 𝑏 = 𝑎 ∗ 𝑐 and 𝑎 ⊕ 𝑏= 𝑎 ⊕ 𝑐 imply 𝑏 = 𝑐
11. Show that every totally ordered set is a lattice.
14. Consultationhours for discussions
For clearing doubts of students consulting time is from 3:10PM to 3:40PM from Monday to
Friday
(M.NARMADHA)

Hand out dm

More Related Content

What's hot

Similar to Hand out dm

Recently uploaded

Hand out dm