This is the appendix in the notes here.
Lists and 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 “” 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 is positive
or numbers such that they are between and and even
The set of numbers greater than zero less than or equal to ten and even. Notice the comma is like an “and”.
A couple of pieces of notation.
- – means “in“ or “belongs to”. E.g. two belongs to the numbers from 1 to 10:
- – 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.
There are some commonly occuring sets with a special notation:
- – the natural numbers,
- – the integers,
- – the rational numbers (aka. fractions),
- – the real numbers, e.g.
- – numbers between and , inclusive and exclusive.
Note that .
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.
We use the symbol 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.)
Normally at school “” is used to mean multiplication. However, people also often use ““. I.e. We use the symbol 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 and ,
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.
A function is something that takes an element from one set and gives you an element from another. E.g. or . We write where is the domain, the set of elements to which we apply the function, and is the range, the set where the function takes its values. In probability we work with the function , i.e. for each outcome in our probability space we assign a probability which is a number between zero and one.
We use the equals sign when two things are the same. I.e. . 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.)
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 gets very close to then gets closer and closer to .
Further the following statement holds: The statement says that as gets larger and larger, the expression gets closer and closer to .