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}.

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