Some Mathematical Notation

Because I heard a few students had not come across some mathematical notation. Here is a quick introduction to a few mathematical terms.

Lists and Sets.

Sets. A set is a list of numbers, letters, objects,.. what ever you want really. We contain these within curly brackets $\{$ and $\}$ . Eg.

$\begin{gathered} \{1,2,3,4,5,6,7,8,9,10\} \\ \{ a, b, c, ..., z \}\, .\end{gathered}$

The order does not matter in a set. For instance,

$\{ 1, 2, 3, 4 \} = \{ 4, 3, 2,1 \} = \{ 2, 4, 1, 3\}\, .$

We ofter refer to the items inside as “elements”.
Sometimes we use dots “ $\dots$ ” when it is clear what is happening next:

$\begin{gathered} \{ a, b, c, ..., z \}\\ \{ 1, 2, 3, ... \} \end{gathered}$

We can use a colon “:“ to specify conditions on a set. We can read this as “such that”. Eg. numbers such that $x$ is positive

$\begin{gathered} \{ x : x > 0 \} \, . \end{gathered}$

or numbers such that they are between $1$ and $10$ and even

$\begin{gathered} \{ x : 0 < x \leq 10 ,\;\, x\text{ even}\} \end{gathered}$

The set of numbers greater than zero less than or equal to ten and even. Notice the comma is like an “and”.

Set Notation. A couple of pieces of notation.

$\in$ – means “in“ or “belongs to”. E.g. two belongs to the numbers from 1 to 10:

$2 \in \{ 1,2,3,4,5,6,7,8,9,10\}\, .$

$\subseteq$ – means “subset”. E.g. the set of number 2,4,6,8,10 is a subset of the numbers from 1 to 10:

$\{ 2,4,6,8,10 \} \subseteq \{1,2,3,4,5,6,7,8,9,10\}\, .$

There are various other notations that I will introduce shortly.

Special sets. There are some commonly occuring sets with a special notation:

$\mathbb N$ – the natural numbers, $\mathbb N = \{1, 2 , 3 ,... \}\, .$
$\mathbb Z$ – the integers, $\mathbb Z = \{ ... , -2, -1, 0, 1, 2, ... \}\, .$
$\mathbb Q$ – the rational numbers (aka. fractions), $\mathbb Q = \Big\{ \frac{a}{b} : a \in \mathbb Z, b\in \mathbb N \Big\}\, .$
$\mathbb R$ – the real numbers, e.g. $\pi \in \mathbb R$
$[a,b], (a,b)$ – numbers between $a$ and $b$ , inclusive and exclusive.

Note that $\mathbb N \subseteq \mathbb Z \subseteq \mathbb Q \subseteq \mathbb R$ .

Ordered lists. Sometimes we want to list elements where the order matters. We contain these with round brackets $($ and $)$ . E.g.

$\begin{gathered} (x,y,z),\\ (\text{heads},\text{tails},\text{heads},\text{tails})\end{gathered}$

(Note this is useful for co-ordinates for geometry but also when we can in what order a sequence of events occur in probability.)

Here the order of elements in these lists does matter:

$( 1, 2, 3, 4 ) \neq ( 4, 3, 2,1 ) \neq ( 2, 4, 1, 3 )\, .$

Again we often use “ $:$ ” to list the items in the list or specify the conditions. E.g. Here we list the probabilities for each outcome from two coin throws.

$( p_\omega : \omega \in \{ HH, HT, TH, HH \} )$

Cardinality of a set. The cardinality of a set is the number of elements in that set. We use brackets $|$ and $|$ to denote the cardinality. Eg.

$| \{a, b, c, d\} | =4 \, .$

Products and Sums.

Sums. We use the symbol $\sum$ for sums over a specified range: $\begin{gathered} 1+2+3+4+5+6+7+8+9 = \sum_{n=1}^9 n,\\ \sum_{\omega \in \Omega} p_{\omega}=1, \\ 1+ \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \sum_{n = 1}^\infty \frac{1}{n^2} \, .\end{gathered}$

Notice sums do not need to be finite. Notice we sum over a range of values in a set. (This is useful in probability.)

Products. Normally at school “ $\times$ ” is used to mean multiplication. However, people also often use “ $\cdot$ “. I.e. $1\cdot 2 \cdot 3\cdot 4 = 24$ We use the symbol $\prod$ for products of a range over values. E.g.

$\begin{gathered} 2^n = \prod_{i=1}^n 2, \\ 1\times 4 \times 6 \times 10 = \prod_{n \in \{\frac{1}{2},2,3,5\} } (2n) \end{gathered}$

Notice that here do products over sets. (This is useful in probability.)

Cartesian Products. We can do products for sets. That is where we create a set consisting of the order pairs from two or more sets.

$\begin{gathered} \{ H, T\} \times \{ H, T \} = \{ (H,H) , (H,T) , (T,H), (T,T) \} \\ \prod_{i=1}^3 \mathbb R = \{ (x,y,z) : x\in \mathbb R , y\in \mathbb R , z\in \mathbb R\}\end{gathered}$ Notice the cardinality a product set is the product of the sizes of the sets:

$\Big| \prod_{i=1}^n \Omega_i \Big| = \prod_{i=1}^n | \Omega_i |$

This is why it makes sense to think of it as a product.

Functions.

A function is something that takes an element from one set and gives you an element from another. E.g. $f(x) = x^2$ or $f(\theta) = e^{i\theta}$ .

We write $f: \mathcal D \rightarrow \mathcal R$ where $\mathcal D$ is the domain, the set of elements to which we apply the function, and $\mathcal R$ is the range, the set where the function takes its values. In probability we work with the function $\mathbb P : \Omega \rightarrow [0,1]$ , i.e. for each outcome in our probability space we assign a probability which is a number between zero and one.