# Set theory

Source: https://projectalevel.co.uk/maths/algebra/set_theory

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**Introduction**
A set is a group of objects. Each object is known as as a member of the set. A set can be represented using curly brackets. So a set containing the numbers 2, 4, 6, 8, 10, ... is: {2, 4, 6, 8, 10, ... } . Sets are often also represented by letters, so this set might be E = {2, 4, 6, 8, 10, ...} . Alternatively, E = {even numbers} .

**Common Sets**
Some sets are commonly used and so have special notation:

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**Other Notation**
If A = {1, 2, 4, 8} . n(A) = 4. This is because n(A) means the number of members in set A. The universal set is the set of all sets. All sets are therefore subsets of the universal set.

**Venn Diagrams**
Venn diagrams are used to represent sets. Here, the set A{1, 2, 4, 8} is shown using a circle. In Venn diagrams, sets are usually represented using circles. The universal set is the rectangle. The set A is therefore a subset of the universal set.

![](https://content.projectalevel.co.uk/img/maths/sets3.gif)

The complement of A, written A', contains all events in the sample space which are not members of A. A and A' together cover every possible eventuality.

![](https://content.projectalevel.co.uk/img/maths/sets4.gif)

AuB means the union of sets A and B and contains all of the elements of both A and B. This can be represented on a Venn Diagram as follows:

![](https://content.projectalevel.co.uk/img/maths/set5.gif)

AnB means the intersection of sets A and B. This contains all of the elements which are in both A and B. AnB is shown on the Venn Diagram below:

![](https://content.projectalevel.co.uk/img/maths/set6.gif)

Clearly n(A) + n(B) - n(AnB) = n(AuB)