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.
- 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”.
Set Notation. 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.
Special sets. 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 .
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 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 “” 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.)
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.
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.
There are various symbols that are used for making logical statements in mathematics. Here are a few:
- – means “for all”.
- – means “there exists”.
- – means “implies”.
- – means “if and only if”.
- s.t. – means “such that”.
- – means “not”.
Eg. For all positive real numbers there exists a natural number, , such that is smaller than .
Eg. For all natural numbers , implies that is divisible by .
Notice how much shorter it is to write the above statements.