CS207 (Discrete Structures)
Exercise problem set 9
Perfect matchings, vertex covers and Stable matchings
October 15, 2015
Solve the following questions from Douglas Wests book, Introduction to
Graph Theory, 2nd Edition.
1. Perfect matchings and vertex co
CS 207: Discrete Structures
Abstract algebra and Number theory
Lagranges theorem and its proof
Lecture 39
Oct 29 2015
1
Relating the size of a group and its subgroups
Theorem (Lagrange)
If H is a subgroup of a nite group G, then H divides G.
2
Relati
CS 207: Discrete Structures
Abstract algebra and Number theory
Cylic groups, Lagranges theorem
Lecture 38
Oct 27 2015
1
Recap
Cyclic group
A group G is cyclic if there exists an element x G such that
every element of G is a power of x. x is called the ge
CS 207: Discrete Structures
Abstract algebra and Number theory
subgroups, cyclic groups, group isomorphisms
Lecture 37
Oct 26 2015
1
Recap
Denition
A group is a set S along with an operator such that:
Closure: a, b S, a b S.
Associativity: a, b, c S, a (
CS 207: Discrete Structures
Abstract algebra and Number theory
Lecture 35
Oct 13 2015
1
Next topic
Abstract algebra and Number theory: An introduction
Denition of a group
Denition
A group is a set S along with a operator such that the
following conditions
CS 207: Discrete Structures
Abstract algebra and Number theory
Lecture 36
Oct 15 2015
1
Last topic of this course
Abstract algebra and Number theory: An introduction
Recall
Denition
A group is a set S along with an operator such that the
following conditi
CS 207: Discrete Structures
Graph theory
Applications of Halls theorem, matchings and vertex covers
Lecture 31
Oct 6 2015
1
Topic 3: Graph theory
Basic denitions and concepts
Characterizations
1. Eulerian graphs: Using degrees of vertices.
2. Bipartite gr
CS 207: Discrete Structures
Abstract algebra and Number theory
Lecture 34
Oct 12 2015
1
Topic 3: Graph theory
Some basic notions
Basics: graphs, paths, cycles, walks, trails, . . .
Cliques and independent sets.
Graph representations, isomorphisms and auto
CS 207: Discrete Structures
Graph theory
Matchings, maximum matchings, augmenting paths
Lecture 29
Oct 1 2015
1
Recap: Matchings
Suppose m people are applying for n dierent jobs. But
not all applicants are qualied for all jobs, and each can
hold at most o
CS 207: Discrete Structures
Graph theory
Bipartite graphs and subgraphs, cliques and independent
sets
Lecture 25
Sept 21 2015
1
Topic 3: Graph theory
Topics covered in the last two lectures:
What is a Graph?
Paths, cycles, walks and trails; connected grap
CS 207: Discrete Structures
Abstract algebra and Number theory
Modular arithmetic and cryptography
Lecture 40
Nov 2 2014
1
Recap: Abstract algebra
Topics covered till now: Summary
Denition of an abstract group; basic properties
Examples:
Invertible matri
CS 207: Discrete Structures
Abstract algebra and Number theory
Modular arithmetic and RSA
Lecture 41
Nov 3 2015
1
More properties of modular arithmetic & group theory
Denition
For integers a, b and positive integer m, if m(a b), then we
say that a is co
CS207 (Discrete Structures)
Exercise problem set 7
September 24, 2015
Solve the following questions from Douglas Wests book, Introduction to
Graph Theory, 2nd Edition.
1. Simple questions: Exercise 1.1.4, 1.1.9
2. Mediumlevel questions: Exercises 1.1.19,
CS207 (Discrete Structures)
Exercise problem set 10 Abstract algebra
October 21, 2015
1. Are the following statements true or false? If true, prove them. If false,
specify which dening property fails as well as give a counterexample. G
be a group and H an
CS207 (Discrete Structures)
Practice problems on graphs, matchings
October 1, 2015
1. Show that in every simple graph there is a path from any vertex of odd
degree to some other vertex of odd degree.
2. A vertex is a cutvertex if its deletion increases t
CS207 (Discrete Structures)
Exercise Problem Set 3
August 8, 2015
Instructions:
Attempt all questions.
If you have any doubts or you nd any typos in the questions, post them on piazza at once!
Relations
1. Give an example for each of the following, if s
CS207 (Discrete Structures)
Exercise Problem Set 2
August 1, 2015
Instructions:
Attempt all questions.
Some of the answers will be discussed during the tutorial sessions, but again you are expected to have
attempted all the questions.
If you have any d
CS207 (Discrete Structures)
Exercise problem set 5
Aug 26 2015
1. Consider the standard deck of 52 playing cards. A balanced hand is a
subset of 13 cards containing four cards of one suit and three cards of each
of the remaining three suits. Find N , the
CS207 (Discrete Structures)
Problem set 4
Aug 21 2015
Attempt all questions.
Apart from things proved in lecture, you cannot assume anything as obvious. Either quote previously proved results or provide clear justication
for each statement.

1
Lattices
CS207 Discrete Structures: Induction, proofs
Exercise Problem Set 1
1. Prove (by induction) or disprove: For every positive integer n,
(a) 12 22 + 32 + (1)n1 n2 = (1)n1 n(n+1) .
2
(b) if h > 1, then 1 + nh (1 + h)n .
(c) 12 divides n4 n2 .
2. Use the well
CS 207: Discrete Structures
Graph theory
Stable matchings
Lecture 3233
Oct 89 2015
1
Topic 3: Graph theory
Topics in Graph theory
1. Basics concepts and denitions.
2. Eulerian graphs: Using degrees of vertices.
3. Bipartite graphs: Using odd length cycl
CS 207: Discrete Structures
Lecture 20 Counting and Combinatorics
PHP and its extensions
Ramsey Theory  A search for order in disorder
Aug 31 2015
1
Recap: Topics in Combinatorics
Counting techniques and applications
1. Basic counting techniques, double
CS 207: Discrete Structures
Lecture 15 Counting and Combinatorics
Solving Recurrence relations and generating functions
Aug 20 2015
1
Find the Fibonacci sequence!
F (n) = F (n 1) + F (n 2).
1, 1, 2, 3, 5, 8, 13, .
2
Find the Fibonacci sequence!
F (n) = F
CS 207: Discrete Structures
Lecture 13 Counting and Combinatorics
Aug 17 2015
1
Last class
Basic counting techniques
Sum and product principles
Bijection principle
Binomial coecients, permutations and combinations
Double counting
2
Last class
Basic counti
CS 207: Discrete Structures
Instructor : S. Akshay
Aug 10, 2015
Lecture 10 Basic mathematical structures: chains, antichains, lattices
1
Recap: Partial order relations
Last class we saw
Partial orders: denition and examples
Posets, chains and antichains