University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Notes and Exercises 3
Introduction to Graph Theory
Walks, Paths, Tours and Cycles
3.1
Basic Denitions
For a set S, use
S
2
to denote the subsets of S with exactly 2 elements; so
 A = 2 . Note that this means that
S
2
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
2013 examination
MA 210
Discrete Mathematics
(Half Unit)
Suitable for all candidates
Instructions to candidates
Time allowed :
2 hours.
This examination paper contains 5 questions. You should answer 4 questions. If
additional questions are answered, only
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
2014 examination
MA 210
Discrete Mathematics
(Half Unit)
Not suitable for resit candidates
Instructions to candidates
Time allowed :
2 hours.
This examination paper contains 5 questions. You should answer 4 questions. If
additional questions are answered
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Summer 2015 examination
MA 210
Discrete Mathematics
(Half Unit)
Suitable for all candidates
Instructions to candidates
This exam contains 5 questions. You may attempt as many questions as you wish, but only
your best 4 questions will count towards the fin
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
Lent 2009
Notes for lectures 5 and 6
2.2
InclusionExclusion Principle (continued)
We want to prove the Inclusion Exclusion Principle:
Theorem 2.4. Let A1 , . . . , An be finite sets. For X cfw_1, . . . , n, define
\
N (X) =
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
2009 examination
MA 210
Discrete Mathematics
(Half Unit)
Instructions to candidates
Time allowed :
2 hours.
This examination paper contains 5 questions. You should answer 4 questions. If
additional questions are answered, only your BEST 4 answers will cou
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
2012 examination
MA 210
Discrete Mathematics
(Half Unit)
Suitable for all candidates
Instructions to candidates
Time allowed :
2 hours.
This examination paper contains 5 questions. You should answer 4 questions. If
additional questions are answered, only
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 6
Exercises 3, Questions 4 13.
4
By examining all the possibilities, we find the following nonisomorphic graphs on 4 vertices:
1.
one graph with 0 edges;
2.
one graph with 1 edg
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 10
Exercises 5, Questions 8 19.
8
To find all codewords x = x1 x2 . . . xn in C, we must solve the equation
x1
x2
H x = 0 , where x =
. .
.
xn
For the given matrix H, this
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
1
(a)
FALSE: By the Handshaking lemma, every graph must have an even number of vertices with
odd degree, which is not the case here.
(b)
FALSE: A Hamilton graph has a cycle through all its vertices, so every vertex must have
degree at least 2, which is no
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 9
Exercises 4, Questions 18;
Exercises 5, Questions 1 7.
Exercises 4
18
(a) This is false, as the following counterexample shows. Let G be the graph with vertex set
cfw_ a, b, c,
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 7
Exercises 3, Question 14;
Exercises 4, Questions 1 8.
Exercises 3
14
(a) Fix any vertex v of Qn . That means that v is represented by a 0,1sequence of length n.
The neighbours
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 8
Exercises 4, Questions 9 17.
9
We will prove that for every n, Kruskals Algorithm will choose edges 12, 13, . . . , 1n. Hence,
the total cost of this minimum cost spanning tree
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
1
(a) (i)
We need to place 13 distinguishable objects (students) into 30 distinguishable places (seats);
this can be done in (30)13 := 30 29 (30 13 + 1) ways.
(a) (ii)
We choose 2 out of the group of 5 students in 52 ways and the we can seat them in 6 sea
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
Lent 2009
Notes for lectures 15 and 16
3
3.1
Introduction to Graph Theory
Basic definitions
Definition 3.1. A graph G = (V ( G ), E( G ) is a set of V ( G ) of vertices together with a set
E( G ) of edges, where E( G ) is a sub
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
Lent 2009
Notes for lectures 3 and 4
1.3
Unordered selections (continued)
Suppose that n distinct objects are given. We have already observed that if we want to
choose r of these objects, the order in which they are chosen is i
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Notes and Exercises 1
Introduction to Counting
Basic Counting Techniques
1.1
Basic Principles of Counting
A nonempty set X has n elements, where n is a natural number, if there exists a bijection
f : X cfw_1, 2, . . .
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Notes and Exercises 4
Trees
Colourings of Graphs
4.1
Trees
Definition 4.1
A tree is a connected graph with no cycles.
A forest is a graph with no cycles. This means that every component of a forest is a tree.
Let G be
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Notes and Exercises 2
Introduction to Recurrence Relations
Generating Functions
2.1
Recurrence Relations
An (innite) sequence is a function f that maps natural numbers (or nonnegative integers)
to the set of real numb
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Notes and Exercises 5
Coding Theory
5.1
Introduction
Coding theory is not the area of mathematics dealing with the theory of secret codes for CIA,
MI5, etc. That subject area is called cryptography.
Coding theory deals
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 4
Exercises 2, Questions 5, 7 12, 13 (a),(b) and 14 (a),(b).
5
(This question was not part of the homework. This solution is included so that you have a
complete set.)
We use ind
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 2
Exercises 1, Questions 11 16 and 18 20
11
One can split an equilateral triangle of sidelength 1 into nine equilateral triangles of sidelength 1/3. By the Pigeonhole Principle,
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 5
Exercises 2, Questions 13 (c), 14 (c) and 15 18;
Exercises 3, Questions 1 3.
Exercises 2
13
(c) In part (b) we found that the generating function of ( an )n0 is the function f
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 3
Exercises 1, Questions 17 and 21 23;
Exercises 2, Questions 1 4 and 6.
17
Solution 1
( x + y + z)10 is the product of 10 times the same factor ( x + y + z). The coefficient of
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
Solutions to Homework Exercises for week 1
Exercises 1, Questions 1 10
1
(a) We have 11 choices for the rst seat. After we made this choice, there are 11 1 = 10
possibilities for the second seat in the row. In general,
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Summer 2010 examination
MA21 0
Discrete Mathematics
Half Unit
2009/2010 syllabus only — not for resit candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains 5 questions. You may attempt as many questions as you wish,
but only your
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
LSE
Summer 2011 examination
MA21O
Discrete Mathematics
Half unit
2010/2011 syllabus only — not for resit candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains 5 questions. You may attempt as many questions as you wish,
but only y
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
Lent 2009
Notes for lectures 15 and 16
3.5
Extremal problems
Let G be a graph. Define the relation on V ( G ) by
u v u = v or there exists a walk in G with u and v as ends.
It is easy to show that is an equivalence relation. Th
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
Discrete Mathematics
MA 210
2015/16
General information
Moodle page:
moodle.lse.ac.uk/course/view.php?id=2148
all kind of information;
copies of notes, exercise sets, solutions;
schedule of topics discussed in lectures;
homework schedule;
old exam pa
University of London The London School of Economics and Political Science
Discrete Mathematics
MATHS MA210

Spring 2016
1
(a)
T
Let A1 , . . . , An be finitePsets. For X cfw_1, . . . , n, define N (X) = iX Ai and, for
i, 1 i n, define i = Xcfw_1,.,n,X=i N (X). Then
A1 A2 An  = 1 2 + 3 4 + + (1)n1 n
n
\
X
X
=
Ai .
(1)i1
i=1 Xcfw_1,.,n,X=i
(1)
iX
(b) (i)
Given that a