MATH1081 Discrete Mathematics
D. Angell/D.Trenerry
4.1 Counting
Counting is probably the earliest mathematical operation you learn, and it
might seem rather strange to have part of a university course devoted to it.
(In fact, it is easy to nd enough mater
MATH1081 Test 2 2009 S1 v2A
February 25, 2015
These solutions were written by Johann Blanco and typed up by Georgia Tsambos. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
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MATH1081 Discrete Mathematics
3.1: Title
MATH1081 Discrete Mathematics
3.2: Introduction
The defining characteristic of modern mathematics is proof .
In ancient times mathematics consisted of a collection of rules
for calculation which seemed to work; it
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
An ordered pair is a collection of two objects in a specified order.
We use round brackets to denote ordered pairs; e.g., (a, b) is an ordered pair.
An ordered pair is a
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
The identity function on a set X is the function iX : X X, iX (x) = x.
The identity function on a set X is the function iX : X X, iX (x) = x.
For any function f : X Y ,
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
Welcome !
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Den
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
Recall that if f is a function from X to Y , then
for every x X, there is exactly one y Y such that f (x) = y.
Recall that if f is a function from X to Y , then
for ever
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
Laws of set algebra:
Laws of set algebra:
Commutative laws A B = B A
AB =BA
Laws of set algebra:
Commutative laws A B = B A
AB =BA
Associative laws
A (B C) = (A B) C
A (
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
Exercise. Given the formulas
n
X
n(n + 1)
k =
2
k=1
and
n
X
n(n + 1)(2n + 1)
k =
,
6
k=1
2
evaluate
10
10
X
X
(k 3)(k + 2) =
(k 2 k 6)
k=1
=
k=1
10
X
k=1
k2
10
X
k=1
k6
You should print the Topic Lecture Notes before the start of each
topic, and bring them along to lectures, together with some extra
notepaper for making handwritten notes.
The pages have useful but not essential colour features, so are best
printed in col
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
Example. The complete bipartite graph Km,n is the simple bipartite graph with
V1 containing m vertices and
V2 containing n vertices;
edges between every vertex in V1 and
UNSW Australia 2016 S1
MATH1081 Discrete Mathematics
Thomas Britz and Dennis Trenerry
MATH1081 Discrete Mathematics
5 Graph Theory
Thanh Tran/Thomas Britz
MATH1081 Discrete Mathematics
Thanh Tran/Thomas Britz
5 Graph Theory
Loosely speaking, a graph is a
MATH1081 Discrete Mathematics
Frances Kuo
3.1 Logic
A proposition (or statement) is a sentence that is unambiguously true or false.
Example.
Canberra is the capital of Australia is a proposition which is true.
1 is a prime number is a proposition which is
MATH1081 Discrete Mathematics
Frances Kuo
3.2 Proofs
The subject matter of this section is mathematical proof itself, and not the
particular results in the lecture notes or tutorial problems.
You learn the techniques and strategies by practice, practice,
MATH1081
Discrete Mathematics
Topic 1
Counting and Cardinality
Dr. Brian Jeeries
School of Mathematics and Statistics
University of New South Wales
Semester 2 2012
BJ (UNSW Maths & Stats)
MATH1081 Discrete Math
Semester 2 2012
1/9
Cardinality
What is coun
MATH1081 Test 1 2010 S1 v1A
February 10, 2015
These answers were written by Johann Blanco and typed up by Georgia Tsambos. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
forums or groups without as
MATH1081 Test 3 2008 S1 v3A
February 10, 2015
These solutions were written by Johann Blanco and typed up by Evgeny Martynov. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
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MATH1081 Test 3 2008 S2 v1A
February 25, 2015
These solutions were written by Johann Blanco and typed up by Evgeny Martynov. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
forums or groups without
MATH1081 Test 3 2009 S1 v2A
February 10, 2015
These solutions were written by Johann Blanco and typed up by Evgeny Martynov. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
forums or groups without
MATH1081 Test 3 2009 S2 v1A
February 16, 2015
These answers were written by Johann Blanco and typed up by Evgeny Martynov. Please be
ethical with this resource. It is for the use of MathSOC members, so do not repost it on other
forums or groups without as
MATH1081
Discrete Mathematics
Topic 2
Integers and Modular Arithmetic
Dr. Brian Jeeries
School of Mathematics and Statistics
University of New South Wales
Semester 2 2012
Factorisation
B.Jeeries
MATH1081 Discrete Mathematics
2 Integers and Modular Arithme
MATH1081 Discrete Mathematics
1 Sets, Functions, and Sequences
A set is a well-defined collection of distinct objects.
An element of a set is any object in the set.
- belongs to or is an element of
/ - does not belong to or is not an element of
F. Kuo/T.
MATH1081
Discrete Mathematics
Topic 1
Functions and Sequences
Dr. Brian Jeeries
School of Mathematics and Statistics
University of New South Wales
Semester 2 2012
Cartesian Product of Sets
An
ordered pair
is a collection of two objects in a specied order.
MATH1081
Discrete Mathematics
Topic 2
Relations
Dr. Brian Jefferies
School of Mathematics and Statistics
University of New South Wales
Semester 2 2012
MATH1081 Discrete Mathematics
B.Jefferies
1 Relations
A relation R from a set A to a set B is a subset o
MATH1081 Discrete Mathematics
3.75: Introduction
3.76: Propositions
Definition. A proposition is a statement which is
unambiguously true or false.
LOGIC and TRUTH TABLES
Logic is the study of how the truth or falsity of a given
statement follows (or not!)
MATH1081 Discrete Mathematics
4.1: Title
MATH1081
DISCRETE MATHEMATICS
MATH1081 Discrete Mathematics
4.2: Introduction
Counting is probably the earliest mathematical operation
you learn, and it might seem rather strange to have part of
a university course
MATH1081 Discrete Mathematics
F. Kuo/T. Britz/D. Chan/D. Trenerry/C. Greenhill
1 Sets, Functions, and Sequences
SETS
A set is a well-defined collection of distinct objects.
An element of a set is any object in the set.
- belongs to or is an element of or