ENGG 2440B: Discrete Mathematics for Engineers
201415 First Term
Homework Set 2 Solutions
Instructor: Anthony Man-Cho So
Teaching Assistant: Yue Man Chung
Problem 1 Observing that S0 = 0, S1 = 1, S2 = 1, S3 = 2, S4 = 2, S5 = 3, . . . , we hypothesize that

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 6
due 11:59pm Thursday December 8
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your collab

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 2
due 11:59pm Thursday October 6
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your collabo

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Homework 6 Solutions
Questions
1. How many 5-card poker hands are there in which:
(a) At least two suits are present in the hand?
Solution: Let A be

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
Lecture 7
2 and 4 November 2015
A rooted tree is a tree together with a designated vertex r called the root. A complete k-ary tree of
depth d is defined recursiv

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
1
Lecture 10
23 and 25 November 2015
The division rule
In how many ways can we place two identical rooks on an 8 by 8 chessboard so that they occupy
different ro

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
Lecture 11
30 November and 2 December 2015
In lecture 4 we learned how to do arithmetic modulo a prime number. We can use modular
arithmetic to evaluate expressi

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
Lecture 8
9 and 11 November 2015
Recurrences are formulas that describe the value of a function of positive integers at an input in
terms of the value of the sam

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
1
Lecture 6
19 and 28 October 2015
Paths and cycles
A path in a graph is a sequence of distinct vertices v0 , . . . , v` such that cfw_vi , vi+1 is an edge in
t

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 1
due 11:59pm Thursday September 22
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your coll

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 4
due 11:59pm Thursday November 10
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your colla

ESTR 2004: Discrete Mathematics for Engineers
201516 First Term
Homework Set 1
Instructor: Anthony Man-Cho So
Due: September 23, 2015
Required Reading: LLM Chapters 3.1 3.4, 9.1 9.2; notes from lectures on September 9,
10, 16, and 17.
SOLVE THE FOLLOWING

ESTR 2004: Discrete Mathematics for Engineers
201516 First Term
Homework Set 2
Instructor: Anthony Man-Cho So
Due: September 30, 2015
Required Reading: LLM Chapters 9.1 9.2, notes from September 16, 17, 23, and 24.
Suggested Reading: GKP Chapters 1.3, 2.1

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Homework 1
due Thursday 22 September
Questions 1 to 6 are worth 10 points each. If you are in ENGG 2440A, please turn in solutions to
four questions

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Homework 3
due Thursday 20 October
Questions 1 to 6 are worth 10 points each. If you are in ENGG 2440A, please turn in solutions to
four questions of

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Homework 4 Solutions
Questions
1. Can the following types of graphs exist?
(a) A graph in which every vertex has degree at least two but has no cycle

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Homework 5 Solutions
1. Find exact closed form expressions for the following sums. Explain how you discovered the
expression and prove that it is cor

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 3
due 11:59pm Thursday October 20
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your collab

CSCI 3130: Formal Languages and Automata Theory
The Chinese University of Hong Kong, Fall 2016
Homework 5
due 11:59pm Thursday November 24
Collaborating on homework is encouraged, but you must write your own solutions in your own
words and list your colla

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
Lecture 3
23 and 30 Sepember 2015
Induction is a method for proving statements of the form For all n, P (n), where n ranges over the
positive integers. It is par

ENGG 2440A: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2015
Lecture 4
5 and 7 October 2015
Number theory is the branch of mathematics that studies properties of the integers. It is full of
conjectures (propositions that h

ENGG 2440B: Discrete Mathematics for Engineers
2016-17 First Term
Solutions to Homework Set 5
Instructor: Anthony Man-Cho So
Teaching Assistant: Pun Yuen Man
Problem 1 (15pts). Every day Mary walks from her home to school, which is located 10 blocks east

ENGG 2440B: Solution for Homework 4
Problem 1:
(a) Choose the bride first, then choose 5 persons from the last 9 persons, and by permutation,
we have 95 6 P6 = 90720 ways to arrange.
(b) Choose the bride or groom first, and then choose 5 persons from the

ENGG2440B: Discrete Mathematics for Engineers
201415 First Term
Homework Set 4 Solutions
Instructor: Anthony Man-Cho So
Teaching Assistant: Zhang Qi
Problem 1
We split Marys walk to school into two parts.
First, Mary needs to pay a visit to her best frien

ENGG 2440B: Discrete Mathematics for Engineers
2016-17 First Term
Solutions to Homework Set 1
Instructor: Anthony Man-Cho So
Teaching Assistant: Conghui Tan
Problem 1 (15pts). Explain what is wrong with the following proof.
Theorem. For all integers n 1,

1
Homework5 Solution
Wei SHI
Problem1.
P
Since the degree of v is the largest, and using the Handshaking Theorem, we get ndeg(v) ni=1 deg(vi ) =
2E. So deg(v) 2E/n.
Problem 2.
The smallest number of vertices in G is k + 1.
Proof: Assume the smallest numbe

ENGG 2440B: Discrete Mathematics for Engineers
2016-17 First Term
Solutions to Homework Set 3
Instructor: Anthony Man-Cho So
Teaching Assistant: Shixiang Chen
Problem 1 (15pts). Let d 1 and a1 , ., ad 0 be given constants. For n 0, define
f (n) =
d
X
ak n

ENGG 2440B: Discrete Mathematics for Engineers
2016-17 First Term
Solutions to Homework Set 2
Instructor: Anthony Man-Cho So
Teaching Assistant: Wang Peng
Problem 1 (40pts). (a) Prove that the following identity holds for all integers n > 0 without using

ENGG 2440B: Discrete Mathematics for Engineers
201415 First Term
Homework Set 3 Solutions
Instructor: Anthony Man-Cho So
Teaching Assistant: Liu Huikang
Problem 1
(a) Since the bride must be in the picture, we firstly choose other 5 people the remained 9

ENGG2440B: Discrete Mathematics for Engineers
201415 First Term
Homework Set 1 Solutions
Instructor: Anthony Man-Cho So
Problem 1
we can see that
Teaching Assistant: Weijie Wu
The theorem has not been proved for n = 2. If we use the method in the proof,
a

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Monday 24 October
Midterm Exam Solutions
1. Is the following deduction rule valid?
xy : P (x, y)
xy : P (x, y)
xy : P (x, y)
Solution: No. Suppose P

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
Lecture 4
The Chinese University of Hong Kong, Fall 2016
28 September and 3 October 2016
Number theory is the branch of mathematics that studies properties of the integers. It is full of
conjectur

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Lecture 5
5 and 12 October 2016
Graphs are an especially popular object of study in discrete mathematics. They represent a finite
collection of objec

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Lecture 2
14 and 19 September 2016
In mathematics, the truth of propositions is established by means of a proof. If there is no proof,
then we do not

ENGG 2440A / ESTR 2004: Discrete Mathematics for Engineers
The Chinese University of Hong Kong, Fall 2016
Lecture 3
21 and 26 September 2016
Induction is a method for proving statements of the form For all n, P (n), where n ranges over the
positive intege