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 g
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
1/22/2015
Ma/CS 6b
Class 8: Planar and Plane Graphs
By Adam Sheffer
The Utilities Problem
Problem. There are three cottages on a
plane and each needs to be connected to
the gas, water, and electricity
MATH 340: Discrete Structures II. Winter 2016.
Due in class on Tuesday, February 16th.
Assignment #2: Planar graphs.
1.
Eulers formula.
a) Let G be a planar graph, such that every vertex of G has degr
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 t
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
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 ver
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 ev
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
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 leas
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
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.
Assignm
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.
Assignm
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 w
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 3
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)
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
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. SubPrim
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,
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 minor
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 an
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,
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 th
Math 340 Solutions to third homework
19 Feb, 2010.
(I will collect assignments from the dropoff box at the math ofce before leaving work on Friday, and I will
leave work some time after 4 PM).
(Diest
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 ver
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 thou
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
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,
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