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
Permuations of size r from n letters. An
ordered arrangement of r objects selected from
a1, . . . , an
is a permutation of n objects taken r at a
time.
Example. Write all the permutations of size
2 from 4 letters a, b, c, d.
Solution.
ab, ac, ad, ba, bc,
Example. Pick a point at random from the
interior of the circle
cfw_(x, y) : x2 + y 2 R2
(radius=R).
(i) What is the sample space ?
Answer:
S = cfw_(x, y) : x2 + y 2 R2
(ii) Write the set of points that are closer to
center than the boundary.
Answer:
E1 =
|E|
3. then P (E) = |S| .
Counting sample points.
Multiplication rule. Suppose that an experiment (procedure) E1 has n1 outcomes and
for each of these possible outcomes an experiment (procedure) E2 has n2 possible outcomes. The composite experiment (proce
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
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
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
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
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
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
Set Theory, Random Experimenets and
Probability
Denition: The sample space S of a random
experiment is the set of all possible outcomes.
Denition: An event E is any subset of the
sample space. We say that E occurs if the
observed outcome x is an element o
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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
Assignment 3: Solutions
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) Eye Witness. Consider a large city w
Math 340: Discrete Structures II
Assignment 5: Solutions
1. Combinatorial Identities.
(a) Give an algebraic proof of the following identity:
2n
2
=2
n
+ n2
2
(b) Give a combinatorial (i.e. bijective) proof.
(c) Give an algebraic proof of the following ide
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
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 |
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 .
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
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
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 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).
(Diestel 5.6) Let G = (V W, E) be the graph with V = cfw_v1 ,
Math 340 Solutions to fourth homework
27 Mar, 2010.
1. (Matousek and Nesetril, Exercise 5.1.2.) We prove the claim of the question by induction on n =
|V |. The base case n = 1 is obvious, as in this case G consists of a single vertex and no edges.
Suppos
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