# A. Mathematical Notation

This is the appendix in the notes here.

## Lists and Sets.

### Sets.

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

• The order does not matter in a set. For instance,
• We ofter refer to the items inside as “elements”.
• Sometimes we use dots “$\dots$” when it is clear what is happening next:
• 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

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

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:

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

(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:

• 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.

### 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.

## Products and Sums.

### Sums.

We use the symbol $\sum$ for sums over a specified range:

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. We use the symbol $\prod$ for products of a range over values. E.g.

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

Recall that for mathematical symbols we often omit the product symbol altogether. E.g. for $x=3$ and $y=5$,

### 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.

Notice the cardinality a product set is the product of the sizes of the sets:

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.

## Equals.

We use the equals sign when two things are the same. I.e. $2 = 6/3$. We also use add a colon “:=” when we are defining something. I.e. the natural numbers are defined by (We could use $=$ here, but, by using $:=$, it makes it more explicit that we are defining a new piece of notation.)

## Limits.

We will occasionally write statements like There is a formal mathematical definition for this, which we do not get into. But it should be reasonably clear that what the above statement is staying is that as $n$ gets very close to $1$ then $1- \frac{1}{n}$ gets closer and closer to $0$.

Further the following statement holds: The statement says that as $n$ gets larger and larger, the expression $(1-x/n)^n$ gets closer and closer to $e^{-x}$.