AST21111 Discrete Mathematics
Tutorial 5 Solution
1. Prove that there are integers m and n such that m > 1 and n > 1 and (1/m + 1/n) is an
integer.
Sol:
For example, let m = n = 2. Then m and n are integers such that m > 0 and n > 0 and (1/m + 1/n)
= 1/2

AST21111
Discrete Mathematics
9. Graph Theory
1
Graphs
Definition:
A graph G consists of a nonempty set of vertices, V(G), and a set of
edges, E(G).
Each edge is associated with one or two vertices called endpoints.
The relationship between an edge and it

AST21111
Discrete Mathematics
2. Logic II
1
What is Logic?
To convince people that your opinion about something is
correct, you need to provide an argument ( ) to support
your claim.
Logic is the study of analyzing and evaluating arguments.
Logic allows y

AST21111
Discrete Mathematics
3. Methods of Mathematical
Proof and Elementary Number
Theory
1
Mathematical Proofs
A mathematical proof is a valid argument that shows the
truth of a mathematical statement is inferred/deduced from
the truth of premises in t

AST21111
Discrete Mathematics
2. Logic II
1
Predicates
In English grammar, the word predicate refers to the
part of a sentence that gives information about the
subject.
In the sentence Vincent Kwan is a teacher at
CCCU. The word Vincent Kwan is the subjec

Strong Mathematical
Induction
A proof by strong mathematical induction also consists
of a basis step and an inductive step.
The basis step may contain proofs for several initial
values.
In the inductive step, the truth of P(n) is assumed not
just for one

AST21111 Discrete
Mathematics
0. Introduction
Teachers
Dr. H. Y. Kwan Vincent (lectures and tutorials)
Email: hykwan@cityu.edu.hk
Tel: 3442-9509
Room: AC2 6420
Dr. K. Y. Kuan, Kevin (tutorials)
Email: kkykuan@cityu.edu.hk
Tel: 3442-7688
Room: AC2 6416
Tea

AST21111
Discrete Mathematics
8. Probability
1
Probability
Probability is a measure of the likelihood of an event in an
experiment where the experiment produces exactly one
out of several possible outcomes.
Definitions:
A sample space is the set of all po

AST21111
Discrete Mathematics
1. Preliminary
1
Sets
A set is a collection of objects.
Those objects are called elements/members of the set.
e.g.
The coins used in Hong Kong forms a set.
The set contains 10, 20, 50, $1, $2, $5 and $10.
20 is a member in th

AST21111
Discrete Mathematics
4. Set Theory
1
Sets
A set is a collection of objects.
Those objects are called elements/members of the set.
A set is denoted by the set-roster notation.
The set-roster notation of a set simply lists all members of
the set in

AST21111
Discrete Mathematics
5. Sequences, Mathematical
Induction and Recursion
1
Sequences
A sequence is an ordered list of numbers, typically it is
written as follows.
Each individual element ak is called a term. The k in ak is
called an index. am is i

AST21111
Discrete Mathematics
6. Functions
1
Functions
Definition:
Let A and B be nonempty sets, a function f from A to B,
denoted by f : A B, (f maps A to B) is an assignment of
each element of A to a unique element of B.
Given a function f : A B and x A

AST21111 Discrete Mathematics
Tutorial 6 Solution
1. Assume that a and b are both integers and that a 0 and b 0. Explain why (b a)/(ab2)
must be a rational number.
Sol:
2. Consider the statement: The cube of any rational number is a rational number.
a. Wr

AST21111 Discrete Mathematics
Tutorial 8 Solution
1. Prove that (A B) (A Bc) = A. (Hint: dividing each sub-proof into two cases)
Sol:
2. Prove that A (A B) = A.
Sol:
3. Prove that for all sets A and B, if A B then Bc Ac.
Sol:
4. Use the set identities in

AST21111 Discrete Mathematics
Tutorial 10 Solution
1. Prove that 2n < (n + 2)! for all integer n 0 by mathematical induction.
2. Prove that n3 > 2n + 1 for all integer n 2 by mathematical induction.
3. Suppose that f0, f1, f2, . . . is a sequence defined

AST21111 Discrete Mathematics
Tutorial 4 Solution
1. Let Q(n) be the predicate n2 30.
a. Write Q(2), Q(2), Q(7), and Q(7), and indicate which of these statements are true and
which are false.
b. Find the truth set of Q(n) if the set of n is Z, the set of

AST21111 Discrete Mathematics
Tutorial 7 Solution
1. Let A = cfw_n Z | n = 5r for some integer r and B = cfw_m Z | m = 20s for some integer s.
a. Is A B? Explain.
b. Is B A? Explain.
Sol:
2. Let A = cfw_1, 3, 5, 7, 9, B = cfw_3, 6, 9, and C = cfw_2, 4, 6,

AST21111 Discrete Mathematics
Tutorial 2 Solution
1. Use Theorem 1.1 in your lecture note to verify the logical equivalences below. Supply a
reason for each step.
a) (p q) (p q) p
b) (p (p q) (p q) p
Sol:
a)
b)
2. Construct a truth table to show pq, its c

AST21111 Discrete Mathematics
Tutorial 1 Solution
1. Let s = stocks are increasing and i = interest rates are steady. Write the following
statements in symbolic form using the symbols , , and and the indicated letters to
represent component statements.
a)

AST21111 Discrete Mathematics
Tutorial 9 Solution
1. Prove that for all integers n 1, 2 + 4 + 6 + + 2n = n2 + n by mathematical induction.
Sol:
2. Prove that 5n 1 is divisible by 4 for each integer n 0 by mathematical induction.
Sol:
3.
Prove that for all

AST21111 Discrete Mathematics
Tutorial 3 Solution
1. Use the logical equivalences to rewrite the statement p q r q without using the
symbol .
Sol:
2. Use truth tables to determine whether the argument forms below are valid. Write some
statements to justif

AST21111 Discrete Mathematics
Tutorial 11 Solution
1. Suppose a sample space S consists of three outcomes: 0, 1, and 2. Let A = cfw_0, B = cfw_1, and C
= cfw_2, and suppose P(A) = 0.4, and P(B) = 0.3. Find each of the following:
a) P(A B), b) P(C) , c) P(

AST21111
Discrete Mathematics
7. Counting
1
The Multiplication Rule
Suppose a computer has four connecting units (A, B, C,
and D) and there are three external devices (X, Y, and
Z). How may ways are there to pair the computer with
only one external device