The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Math2343: Problem Set 2
(Deadline: 7 Oct. 2011)
1. Show that the set Q of rational numbers is countable.
2. Let be a countable set. Show that is countable.
3. Let B = cfw_0, 1. Show that the set B is uncountable.
4. Find a onetoone correspondence betwee
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Graph Theory
1
Graphs and Subgraphs
Definition 1.1. A multigraph or just graph is an ordered pair G = (V, E)
consisting of a nonempty vertex set V of vertices and an edge set E of
edges such that each edge x 2 E is assigned to an unordered pair cfw_u, v w
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Binary Relations
1
Binary Relations
The concept of relation is common in daily life and seems intuitively clear. For
instance, let X be the set of all living human females and Y the set of all living
human males. The wifehusband relation R can be thought
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Fall 2011
Math 2343 Discrete Structure
Brief Outline: This is an introductory course on Discrete Mathematics for Year One
students. We will cover set theory, elementary logic, binary relations, combinatorics,
graph theory, and Boolean algebra, etc. If tim
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
1
Divisibility
Given two integers a, b with a 6= 0. We say that a divides
b, written
a  b,
if there exists an integer q such that
b = qa.
When this is true, we say that a is a factor (or divisor) of
b, and b is a multiple of a. If a is not a factor of b,
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Math132: Problem Set 4
(Deadline: Friday, 29 Oct. 2004)
1. A computer user name consists of three English letters followed by five digits. How many different user
names can be made?
2. A set lunch includes a soup, a main course, and a drink. Suppose a cus
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Math132: Problem Set 3
(Deadline: Friday, 15 Oct. 2004)
1. (a) Verify the following
25 67 (mod 21),
3 25 3 67 (mod 21),
3 14 3 28 (mod 21),
14 6 28 (mod 21).
(b) Let a be an integer. Verify that for any integers k, l Z,
a + kn a + ln (mod n).
(c) Find all
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Counting
1
Counting Principle
Let X, Y be finite sets. If X, Y are disjoint, then
X [ Y  = X + Y .
For two tasks T1 and T2 to be performed in sequence, if the task T1 can be performed
in m ways, and for each of these m ways the task T2 can be perfor
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
1
Divisibility
Given two integers a, b with a 6= 0. We say that a divides
b, written
a  b,
if there exists an integer q such that
b = qa.
When this is true, we say that a is a factor (or divisor) of
b, and b is a multiple of a. If a is not a factor of b,
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
1
Mathematical Induction
We assume that the set Z of integers are well defined, and
we are familiar with the addition, subtraction, multiplication,
and division. In particular, we assume the following axiom for
subsets of integers bounded below.
WellOrde
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Recurrence Relations
1
Infinite Sequences
An infinite sequence is a function from the set of positive integers to the set of
real numbers or to the set of complex numbers.
Example 1.1. The game of Hanoi Tower is to play with a set of disks of
graduated si
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Week 12
1
Some Warmup Questions
Abstraction: The process going from specific cases to general problem.
Proof: A sequence of arguments to show certain conclusion to be true.
If . then .: The part after if is called the hypothesis, the part after
then is
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Week 34
1
Statements
A mathematical statement or sentence (or proposition) is a declarative
sentence that is either true or false, but not both. The truth value (true, false)
for any statement can be determined and is not ambiguous in any sense. The
true
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
Math2343: Problem Set 1
(Deadline: 19 Sept. 2011)
1. Let Z be the universal set. Let A be the set of integers whose elements are multiples of 3, and let B
A B,
be the set of integers whose elements are the multiples of 5. Find the sets A B, A B, A,
and A
The Hong Kong University of Science and Technology
Discrete Structure
MATH MATH 2343

Spring 2012
(CHEM1030)[2013](s)final~=fn7epee^_35131.pdf downloaded by thwongau from http:/petergao.net/ustpastpaper/down.php?course=CHEM1030&id=0 at 20160810 08:41:40. Academic use within HKUST only.
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