Math 340: Discrete Structures II
Assignment 1: Solutions
1. Stable Marriages. As we illustrate the algorithms solution we will highlight current
engagements in bold. The algorithm ends as soon as we get ve engagements. In
each step, the lowest index singl
Math 340: Discrete Structures II
Due: Friday, January 30th.
Assignment 1: Matchings
You may work with a couple of friends. As a last resort, you may even use
some other written resources but you must cite them. In all cases you must
write up your own solu
Math 340: Discrete Structures II
Due: Friday, January 30th.
Assignment 1: Matchings
You may work with a couple of friends. As a last resort, you may even use
some other written resources but you must cite them. In all cases you must
write up your own solu
MATH 340: Discrete Structures II. Winter 2017.
Assignment #3: Discrete Probability.
1.
Bayes Theorem.
a) Babies. A family has two children, and at least one of them is a boy.
What is the probability that the family has one boy and one girl?
b) A die and a
MATH 340: PROBABILITY REVIEW.
Basic notions. A sample space S is a finite set, considered as a set of possible outcomes of
an experiment.
An event is a subset of a sample space.
A function p : S R is a probability distribution on S if
p(x)
P 0 for every
MATH 340: GRAPH THEORY REVIEW.
Basic notions. Review of MATH 240.
Definition of a graph. A graph G is an ordered pair (V (G), E(G), where V (G) is a set
of vertices, E(G) is a set of some pairs of vertices called edges.1 We will write uv, instead of
cfw_u
MATH 340: ENUMERATION REVIEW.
Counting using bijections
Review of MATH 240. A function f : X Y is
a surjection (or onto) if for every y Y there exists x X such that y = f (x),
an injection if for every y Y there exists at most one x X such that y = f (x
MATH 340: Discrete Structures II. Winter 2017.
Assignment #1: Matchings.
1.
Stable matching algorithm. Apply the Boy Proposal algorithm to
find a stable matching given the preference lists below. Are there any other
stable matchings?
B1
B2
B3
B4
B5
: G3
:
MATH 340: Discrete Structures II. Winter 2017.
Assignment #2: Planar graphs. Solutions.
1.
Eulers formula.
a) Let G be a planar graph, such that every vertex of G has degree at
least five, and at least one vertex of G has degree eight. Show that G
has at
Math 340: Discrete Structures II
Due: Monday February 16th.
Hand in to my oce (Burnside Hall 1113) by 1PM.
Same rules as assignment 1 apply about working together or using any
outside sources.
Assignment 2: Colourings and Planarity
1. Market-Clearing Pric
Math 340: Discrete Structures II
Due: Monday February 16th.
Hand in to my oce (Burnside Hall 1113) by 1PM.
Same rules as assignment 1 apply about working together or using any
outside sources.
Assignment 2: Colourings and Planarity
1. Market-Clearing Pric
MATH 340: Discrete Structures II. Winter 2017.
Assignment #6: Generating functions. Solutions.
1.
Fruit salad. Let s(n) be the number of ways to make a fruit salad
with n pieces of fruit, given that we must use strawberries by the halfdozen, an even numbe
MATH 340: Discrete Structures II. Winter 2017.
Assignment #4: Discrete Probability II.
1.
The Birthday problem. Suppose that the birthdays of n people in the
room are uniformly distributed among the 365 days of the year. Estimate
how large should n be to
MATH 340: Discrete Structures II. Winter 2017.
Assignment #5: Enumeration. Solutions.
1.
Combinatorial identities.
a) Give an algebraic proof of the following identity:
(
)
n ( )
n+1
k
=
m+1
m
k=m
b) Give a combinatorial (bijective) proof of the identity
McGill University
Department of Mathematics and Statistics
MATH 340 Discrete Structures II. Winter 17
Instructor: Sergey Norin
E-mail: [email protected]
Web: http:/www.math.mcgill.ca/snorin/math340W17.html
Office: Burnside 1116
Office Phone: (514) 398-381
Math 340: Discrete Structures II
Assignment 5: Combinatorics
Questions are worth 10 marks unless otherwise stated. Due: Friday, April 17, Noon. Under my door, or at the Math/Stats Dept.
Oce
1. SubPrime Crisis (taking from Exercise 19.26 in Lehman and Leig
Math 340: Discrete Structures II Due: Monday. March 30th, 4PM (in my oce)
Assignment 4: Probability
Questions are worth 10 marks unless otherwise stated.
1. Generalized Bayes Theorem.
Let Ei : i = 1, 2 . . . n be disjoint (mutually exclusive) events which
Math 340: Discrete Structures II
Due: Feb 27th, 1PM after midterm
Assignment 3: Minors, Connectivity and Flows
Questions are worth 10 marks unless otherwise stated.
1. Minors.
Find the following minors explicitly by saying which edges/vertices you delete
February 14, 2013, 8:35Am - 9:50 AM.
Math 340: Discrete Structures II
Midterm Exam MATH 340
Instructions. The exam is 75 minutes long. The exam contains 4 questions, each worth 10
marks. Write your answers clearly in the notebook provided, showing your wo
Math 340: Discrete Structures II
February 27, 2015, 11:35AM - 12:55 PM.
Midterm Exam MATH 340
This exam contains 4 questions, each worth 10 marks. Write your answers clearly in the
notebook provided, showing your work. You may quote any result/theorem see
MATH 340: Discrete Structures II. Winter 2017.
Due in class on Wednesday, March 8th.
Assignment #3: Discrete Probability.
1.
Bayes Theorem.
a) Babies. A family has two children, and at least one of them is a boy.
What is the probability that the family ha
MATH 340: Discrete Structures II. Winter 2017.
Due in class on Friday, January 27th.
Assignment #1: Matchings.
1.
Stable matching algorithm. Apply the Boy Proposal algorithm to
find a stable matching given the preference lists below. Are there any other
s
Math 340 Solutions to midterm
March 8, 2010
1. (a) Halls theorem states that if G = (V, E) is a bipartite graph with bipartition V = V1 V2 and
|V1 | |V2 |, then there exists a matching hitting all vertices in V1 if and only if for all S V1 ,
|N (S)| |S|.
Math 340 Solutions to fth homework
9 Apr, 2010.
1. (Matousek and Nesetril, Exercise 10.1.3.) Ill answer a more general question. Suppose we start with
n coins, c1 , . . . , cn . A weighing can be thought of as a partition of the coins into sets W , Wr , W
Solutions second homework
8 Feb, 2010
(1 (a) We assume k 1 or else part 1 (a) is false.
Let W be a directed walk starting at s and using only edges of E , with no repeated edges, and of maximal
length subject to this. That is to say, there does not exist
Solutions rst homework
22 January, 2010
1. Let M be a matching in G and write M = cfw_u1 , v1 , cfw_u2 , v2 , . . . cfw_uk , vk , so |M | = k. If S is a minimum
vertex cover of G then for each i = 1, 2, . . . , k, the set S must contain either ui or vi .
Math 340: Discrete Structures II
Assignment 4: Solutions
1. Random Walks. Consider a random walk on an connected, non-bipartite, undirected graph G. Show that, in the long run, the walk will traverse each edge
with equal probability.
Solution. Recall from
Math 340: Discrete Structures II
Midterm Exam : Solutions
1. Matchings. Take a bipartite graph G = (V, E) where the two parts of V in the bipartition are X and Y , where |X| = |Y | = n.
(a) State Halls Theorem.
A bipartite graph G = (V, E) with |X| = |Y |
Math 340: Discrete Structures II
Assignment 2: Solutions
1. Permutation Matrices. The rst thing to notice here is that a permutation
matrix can be identied with a perfect matching between rows and columns.
A 1 in entry Pi,j of a permutation matrix P can b