MATH 222 FALL 2012, Assignment One
(Answer key)
1. (a) Draw a 6-vertex graph G which has two vertices of degree i for each i = 1, 2, 3.
(b) Construct a 10-vertex graph H from the graph in (a) by adding four new vertices and
some edges (incident with the n
201309 Math 222 Assignment 3 Solution Ideas
1. (5 marks) How many sequences of n letters chosen from cfw_A, B, . . . , Z have the property that the letters are in non-decreasing alphabetical order, or in non-increasing
alphabetical order?
Let xA be the nu
CHAPTER 6. RECURRENCE RELATIONS
99
Week 11
Recurrence Relations (Continued)
Nonhomogeneous Recurrences
3. The Towers of Hanoi, also called the Towers of Brahma, is a mathematical game or
puzzle. It consists of three rods, and a number of disks of dierent
MATH 222: Assignment 3
Due: Friday, June 21, 2013
Assignments are due in class before the start of lecture. Late assignments may be deducted
marks and no assignments will be accepted after the end of the lecture.
All answers must include full explanations
MATH 222 FALL 2011, Assignment One
(Answer key)
1. Let f : N N be dened by f = cfw_(x, 3x) : x N. Let S be the set of all even
natural numbers. Find the following sets.
(a) f (S ).
f (S ) = cfw_ 6 k : k N .
(b) f 1 (S ).
f 1 (S ) = S .
(c) f 1 (2011).
f
MATH 222 FALL 2012, Assignment Seven
(Answer key)
1. For each of the following functions, nd the sequence which is generated by the function.
(a)
(13x)3
(1+4x)4 ;
(1 3x)3
=
(1 + 4x)4
3
i
(3x)
i=0
i=0
i+3
(4x)i
i
generates the sequence
3
(3)i
i=0
(b)
ki+3
201001 Math 222 Practice Midterm 1 Questions
Most of these are from actual old midterms or nals.
The rest come from my imagination.
The actual midterm will not be this long.
1. Determine the number of integers between 10000 and 99999 with
(a) exactly thre
MATH 222 FALL 2012, Assignment Six
(Answer key)
1. Let S be the set of all subset of cfw_1, 2, 3, 4, 5. Dene a relation on S by
X RY X cfw_1, 2, 3 = Y cfw_1, 2, 3, X, Y S.
(a) Verify that R is an equivalence relation on S
We show that R is reexive, symmet
CHAPTER 5. GENERATING FUNCTIONS
89
Week 10
Calculating Techniques (continued)
Example 4
How many ways can two dozen identical robots be assigned to four assembly lines so that at
least three but no more than nine robots are assigned to each line?
The gene
CHAPTER 1. GRAPH THEORY
1.7
38
Planar Graphs
Week 4
Planar Graphs
Denition 1.14 Memorize! A graph is planar if it can be drawn in the plane so that
edges intersect only at vertices, otherwise it is called nonplanar.
Such a drawing is called an embedding o
CHAPTER 3. INCLUSION AND EXCLUSION
79
Week 9
If an -set is randomly permuted, all permutations are equally likely, so the probability of a
derangement is
() X
=
(1) ! by Theorem 2.1.4.
!
=0
Recall the Maclaurin series for :
X
2 3
+
+ =
!
=1+
2!
3!
=0
At
Chapter 0
Preliminaries
Week 1
0.1
Sets
We make use of the notation, terminology and concepts developed in MATH 122. We begin by
revising the notation for the most important subsets of the set of real numbers:
N
natural numbers 1 2
integers
Z
rational
CHAPTER 1. GRAPH THEORY
17
Week 2
1.2
Denitions and Basic Properties
Recall the denition of a graph on p. 9 (Week 2). We now dene a number of other basic graph
theoretic concepts. Memorize them!
Graphs
Denition 1.1 A (simple) graph = ( ) consists of a nit
CHAPTER 2. COUNTING: FUNDAMENTAL TOPICS
61
Week 7
2.4
Combinations with Repetitions
Example 1
Seven students go to a burger joint where each buys a beef, fish, chicken or veggie burger.
How many dierent purchases are possible? Some possible choices are li
CHAPTER 2. COUNTING: FUNDAMENTAL TOPICS
68
Week 8
The Pigeonhole Principle (Continued)
6. Show that any 6-element subset of = cfw_1 2 9 contains two elements whose sum is
10.
Here the pigeons and pigeonholes are not immediately obvious.
But let us begin
MATH 222 [A01] Spring 2012
Second Midterm Test
Instructor: Dr. Jing Huang
March 23, 2012
12:30 - 1:20pm
Instructions: This question paper has ve pages plus cover. For each
question, write out your solution/answer carefully in the space provided.
Use the b
The Pigeonhole Principle
Recall that a function f : A B is one-to-one (or 1-1) if dierent elements
of A have dierent images in B . For such a function to exist, it is necessary that
|B | |A|. If |A| > |B |, then no function f : A B can be 1-1 and, thus, s
201309 Math 222 Assignment 5
Due: Wednesday, December 4, 2013, in class, before the lecture starts
Marks for each question as as shown. A total of 36 marks, plus 10 bonus marks, are available
on the two pages. Answers that are complete and correct will re
201309 Math 222 Assignment 4 Solution Ideas
1. (3 marks) Let R be an relation on the non-empty set A. Let DR be the directed graph
with vertex set A, and an arc from x to y if and only if (x, y) R. Describe D if R is
an equivalence relation on A. That is,
201309 Math 222 Assignment 1 Solution Ideas
1. (a)
n
2
|EG |
(b) (n 1) k
(c) Yes. For example when G has vertex set cfw_x, y and edge set cfw_xy. It isnt possible
if G has at least 5 vertices. Suppose G has bipartition (X, Y ). Then either X or
Y contain
201309 Math 222 Assignment 2 Solution Ideas
1. In this question we will use to use strong induction on the number of edges prove that
if G is connected and every vertex has even degree, then G has an Eulerian circuit.
(a) (1 mark) Verify the statement for
MATH 222 FALL 2015, Assignment One
(due Friday Sep. 25 in class before the lecture begins)
Show yourwork clearly. Illegible or disorganized solutions will receive no credit.
1. (a) Draw a 6-vertex graph G which has two vertices of degree i for each i =
1,
201001 Math 222 Sample Final Exam Solution Ideas
1. (a)
7!
2!3!2!
(b) Let xi be the number of times the digit i appears. Then we want the number of
integer solutions to x0 + x1 + . . . + x9 = 7 subject to xi 0, 0 i 9, which
equals 7+91 .
7
2. The LHS coun
UNIVERSITY OF VICTORIA
APRIL EXAMINATIONS 2010
MATH 222: Discrete and Combinatorial Mathematics
CRN: 24272
INSTRUCTOR: G. MacGillivray
NAME:
V00#:
Duration: 3 Hours.
Answers should be written on the exam paper.
The exam consists of 20 questions, for a tot
201001 Math 222 Sample Final Exam
1. Let S1 and S2 be sequences of 7 decimal digits. Suppose we say that S1 is equivalent
to S2 if S1 is a rearrangement of S2 . For example 1231236 is equivalent to 2263311.
(a) How many sequences are equivalent to 9955500
Chapter 2
Counting: Fundamental Topics
Week 6
2.1
The Rules of Sum and Product
Sum and Product Rules (Review)
Suppose the set can be expressed as the union of two nonempty disjoint sets, i.e. = 1 2 ,
where 1 6= 6= 2 and 1 2 = .
How do we nd |1 2 | if |1
CHAPTER 1. GRAPH THEORY
44
Week 5
Graph Colouring Continued
We can also characterize graphs with = 2.
Fact 1.14 A graph satises () = 2 if and only if is bipartite and has at least one edge.
Proof. Suppose is bipartite with bipartition cfw_1 2 , and has at
CHAPTER 1. GRAPH THEORY
28
Week 3
Regular Graphs and Complements
Denition A graph is called -regular , or simply regular if is not important, if
deg = for each ().
For example, the complete graph is 1-regular, while the complete bipartite graph
is -regul