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1.3 Preliminaries

In this section we carry out some basic preparatory work and establish notation which will be used throughout the rest of this module (and in many subsequent ones).

In many ways, mathematics is like a language; an important part of doing mathematics at university level is learning to read and write it correctly. We need to learn both which symbols to use and how to put them together to communicate what we want to say in a way that should be clear to any reader.

We begin with a concept even more basic than that of a number.

Definition 1.3.1

A set is a collection of objects. The objects concerned are called the elements (or members) of the set.

Sets are usually denoted by capital letters $A$ , $B$ and so on. The elements of a set may be anything, but most of the sets we shall consider will be collections of numbers.

Notation 1.3.2

We use the following symbols for particular sets of numbers.

$\mathbb{N}\phantom{{}_{0}}$

(written $\mathbb{N}$ ) is the set of natural numbers: $1,2,3,\dots$
$\mathbb{N}_{0}$

(written $\mathbb{N}_{0}$ ) is the set of natural numbers and zero: $0,1,2,3,\dots$
$\mathbb{Z}\phantom{{}_{0}}$

(written $\mathbb{Z}$ ) is the set of integers: $0,\pm 1,\pm 2,\pm 3,\dots$
$\mathbb{Q}\phantom{{}_{0}}$

(written $\mathbb{Q}$ ) is the set of rational numbers: those that can be written in the form $a/b$ , where $a$ and $b$ are integers (and $b\neq 0$ ).
$\mathbb{R}\phantom{{}_{0}}$

(written $\mathbb{R}$ ) is the set of real numbers: all ‘‘points on the number line’’, or equivalently all numbers having a (possibly infinite) decimal expansion — e.g., $\sqrt{2}$ , $\pi$ , $e$ and $1.57$ .

(For now we shall simply assume that we know what the elements of these sets are; later we shall consider this question in a little more detail.)

Notation 1.3.3

Let $A$ be a set. We write $x\in A$ to mean ‘‘ $x$ is an element of $A$ ’’ (or ‘‘ $x$ belongs to $A$ ’’). Similarly, we write $x\not\in A$ to mean ‘‘ $x$ is not an element of $A$ ’’ (or ‘‘ $x$ does not belong to $A$ ’’). These can also be written the other way round: for example, $A\ni x$ means ‘‘ $A$ contains $x$ ’’, which is the same as $x\in A$ .

Example 1.3.4

We have $-1\in\mathbb{Z}$ but $-1\not\in\mathbb{N}$ ; also $\mathbb{R}\ni\sqrt{2}$ , but $\mathbb{Q}\not\ni\sqrt{2}$ .

There is one particular set which is distinguished by having no elements at all!

Definition 1.3.5

The set containing no elements is called the empty set, and is written $\emptyset$ ; any set other than the empty set is called non-empty.

We shall next introduce two different methods to link a set with all of its elements.

Notation 1.3.6

(i)

We can describe a finite set simply by listing all of its elements, using curly brackets to enclose them; for example, the set whose elements are $1$ , $2$ and $3$ is written $\{1,2,3\}.$

This notation may be extended to infinite sets by using dots (provided that it is clear what is meant); thus we may write

$\mathbb{N}=\{1,2,3,\dots\}\qquad\text{and}\qquad\mathbb{Z}=\{\dots,-2,-1,0,1,2% ,\dots\}.$
(ii)

When listing the elements of a set is not possible, we may instead describe it by specifying a defining property. More precisely, if $A$ is a set and $P(x)$ stands for some statement involving $x$ , then we write

$\bigl\{x\in A:P(x)\bigr\}$

for ‘‘the set of all $x$ in $A$ such that $P(x)$ is true’’ (the colon is read ‘‘such that’’). Thus for example

$\{x\in\mathbb{R}:x^{2}-6x+8=0\}$

is the set of real solutions to the equation $x^{2}-6x+8=0$ , i.e., $\{2,4\}$ . Similarly, we have

$\{n\in\mathbb{N}:n\hbox{ is even}\}=\{2,4,6,\dots\}.$

This set could also be specified as $\{2m:m\in\mathbb{N}\}$ , using a variant of this notation. Another example of this variant is

$\{n^{2}:n\in\mathbb{N}\}=\{1,4,9,16,\dots\}.$

Example 1.3.7

(i)

$\{n+3:n\in\mathbb{N}\}=\{4,5,6,7,\dots\}.$
(ii)

$\{n:n+3\in\mathbb{N}\}=\{-2,-1,0,1,\dots\}.$

Note the difference between (i) and (ii): what goes before the colon gives the elements themselves, while what goes after the colon gives any conditions they must satisfy.

(i)

$\{x\geqslant 0:x^{2}=1\}=\{1\}.$
(ii)

$\{x+1:x^{2}=1\}=\{x+1:x=\pm 1\}=\{0,2\}.$

Finally we state some definitions involving pairs of sets; they may well be familiar.

Definition 1.3.8

Given two sets $A$ and $B$ , we say that $A$ is a subset of $B$ if every element of $A$ also lies in $B$ ; we write $A\subseteq B$ (or $B\supseteq A$ ) to indicate this.

Example 1.3.9

$\{1,2,3\}\subseteq\{1,2,3,4,5,6\}$ and $\{1,2,3\}\supseteq\{1,2\}.$

Clearly if $A\subseteq B$ and $B\subseteq A$ then $A=B$ , since $A$ and $B$ have the same elements.

The remaining definitions provide ways of constructing a new set from two old ones.

Definition 1.3.10

Given two sets $A$ and $B$ , their intersection is the set of elements common to both $A$ and $B$ ; it is written $A\cap B$ . If $A\cap B=\emptyset$ , then we say that $A$ and $B$ are disjoint.

Thus we have $A\cap B=\{x:x\in A\hbox{ and }x\in B\}$ . Clearly $A\cap B\subseteq A$ and $A\cap B\subseteq B$ .

Example 1.3.11

$\{1,2,3,4\}\cap\{3,4,5,6\}=\{3,4\}$ and $\{1,2\}\cap\{3,4\}=\emptyset.$

Definition 1.3.12

Given two sets $A$ and $B$ , their union is the set of elements which lie in at least one of $A$ and $B$ ; it is written $A\cup B$ .

Thus we have $A\cup B=\{x:x\in A\hbox{ or }x\in B\}$ . Clearly $A\subseteq A\cup B$ and $B\subseteq A\cup B$ .

Example 1.3.13

$\{1,2,3,4\}\cup\{3,4,5,6\}=\{1,2,3,4,5,6\}$
and $\{1,2\}\cup\{3,4\}=\{1,2,3,4\}.$

Definition 1.3.14

Given two sets $A$ and $B$ , their (set-theoretic) difference is the set of elements which belong to $A$ , but not to $B$ ; it is written $A\setminus B$ .

Thus we have $A\setminus B=\{x\in A:x\notin B\}$ .

Example 1.3.15

(i)

$\{1,2,3,4\}\setminus\{2,4,6\}=\{1,3\}$ and $\{2,4,6\}\setminus\{1,2,3,4\}=\{6\}$ ;
(ii)

$\mathbb{N}_{0}\setminus\mathbb{N}=\{0\}$ and $\mathbb{N}\setminus\mathbb{N}_{0}=\emptyset$ .