Unit 1
Mathematical Preliminaries
1.1 INTRODUCTION
This unit contains a brief summary of concepts, definitions, notation, and results that you are
assumed to be familiar with.
1.2 SETS
We assume that you are familiar with the following sets of numbers.
R, the set of real numbers.
Z, the set of integers.
Q, the set of rational numbers.
We shall use the above notation throughout.
To indicate that x is an element of a set S, i.e. that S contains x, we write
x 2 S:
To denote that x is not an element of S, we write
= S:
x2
A set S is usually specified in one of two ways. One way is by listing its elements. We do this listing
by writing the elements that belong to the set between curly brackets. For example, if A is the set elements
consisting only of the four elements 2; 4; 6 and 8 we write
A D f2; 4; 6; 8g :
If a set has infinitely many elements, we cannot list them all, but sometimes it is clear what set we
have in mind if we list only a few of its elements. For example, if we write
B D f2; 4; 6; 8; : : :g ;
it is clear that we mean that B is the set of all positive even numbers.
1
, 2
The other way of specifying a set involves set-builder notation. If S is the set of all elements in
set-builder some universal set U that have property P; then we write
notation
S D fx 2 U j x has property Pg :
For example, instead of writing B D f2; 4; 6; 8; : : :g we can write
B D fx 2 Z j x > 0 and x is eveng:
equality Two sets A and B are said to be equal, written A D B, if they have precisely the same elements.
of sets The order in which the elements are listed is not important, for example fa; b; cg is precisely the
same set as fb; a; cg :
subset A is a subset of B, written A B, if every element of A is an element of B.
Clearly, A D B if and only if A B and B A:
proper If A B but A 6D B, we write A B and we say A is a proper subset of B.
subset
An important proper subset of R is the set of positive real numbers, which we denote by RC :
Thus
RC D fx 2 R j x > 0g :
Similarly, we denote the set of positive integers by ZC .
It is also convenient to consider a set with no elements. Such a set is called the empty set, and is
empty set usually denoted by ;. The empty set is a subset of every set.
We now define three operations involving sets, namely [, \ and :
Definitions 1.2.1 Let A and B be two subsets of a universal set U:
1. The union of A and B; denoted by A [ B is the set consisting of all elements that belong to
A or to B, i.e.
A [ B D fx 2 U j x 2 A or x 2 Bg :
2. The intersection of A and B, denoted by A \ B is the set consisting of all elements that
belong to both A and B, i.e.
A \ B D fx 2 U j x 2 A and x 2 Bg :
3. The complement of A relative to B, written B A, consists of all elements in B which are
not in A; i.e.
B = Ag:
A D fx 2 B j x 2
Mathematical Preliminaries
1.1 INTRODUCTION
This unit contains a brief summary of concepts, definitions, notation, and results that you are
assumed to be familiar with.
1.2 SETS
We assume that you are familiar with the following sets of numbers.
R, the set of real numbers.
Z, the set of integers.
Q, the set of rational numbers.
We shall use the above notation throughout.
To indicate that x is an element of a set S, i.e. that S contains x, we write
x 2 S:
To denote that x is not an element of S, we write
= S:
x2
A set S is usually specified in one of two ways. One way is by listing its elements. We do this listing
by writing the elements that belong to the set between curly brackets. For example, if A is the set elements
consisting only of the four elements 2; 4; 6 and 8 we write
A D f2; 4; 6; 8g :
If a set has infinitely many elements, we cannot list them all, but sometimes it is clear what set we
have in mind if we list only a few of its elements. For example, if we write
B D f2; 4; 6; 8; : : :g ;
it is clear that we mean that B is the set of all positive even numbers.
1
, 2
The other way of specifying a set involves set-builder notation. If S is the set of all elements in
set-builder some universal set U that have property P; then we write
notation
S D fx 2 U j x has property Pg :
For example, instead of writing B D f2; 4; 6; 8; : : :g we can write
B D fx 2 Z j x > 0 and x is eveng:
equality Two sets A and B are said to be equal, written A D B, if they have precisely the same elements.
of sets The order in which the elements are listed is not important, for example fa; b; cg is precisely the
same set as fb; a; cg :
subset A is a subset of B, written A B, if every element of A is an element of B.
Clearly, A D B if and only if A B and B A:
proper If A B but A 6D B, we write A B and we say A is a proper subset of B.
subset
An important proper subset of R is the set of positive real numbers, which we denote by RC :
Thus
RC D fx 2 R j x > 0g :
Similarly, we denote the set of positive integers by ZC .
It is also convenient to consider a set with no elements. Such a set is called the empty set, and is
empty set usually denoted by ;. The empty set is a subset of every set.
We now define three operations involving sets, namely [, \ and :
Definitions 1.2.1 Let A and B be two subsets of a universal set U:
1. The union of A and B; denoted by A [ B is the set consisting of all elements that belong to
A or to B, i.e.
A [ B D fx 2 U j x 2 A or x 2 Bg :
2. The intersection of A and B, denoted by A \ B is the set consisting of all elements that
belong to both A and B, i.e.
A \ B D fx 2 U j x 2 A and x 2 Bg :
3. The complement of A relative to B, written B A, consists of all elements in B which are
not in A; i.e.
B = Ag:
A D fx 2 B j x 2
