# Venn diagrams

## In a nutshell

A Venn diagram is a visual display to represent overlap of elements of sets. Each set of a Venn diagram is represented by a closed "bubble" or circle, and the elements of the set are displayed within the bubble. This makes it easy to visualise which data belongs to multiple sets at the same time, and also which data belongs to no sets.

## Sets

### Definition

A set is a collection of items.

The data contained by a set are elements or members of the set, and these members may belong to multiple sets.

This data can be anything, and need not be of the same type. The elements of a set are written in curly brackets and if a set is empty this is signified by either $\{\}$ or $\emptyset$. Typically capital letters are used to indicate sets and lowercase letters are used to indicate elements of sets.

##### Example 1

$A = \{ 1, 5, 7 \}, B =\{ 2,3,\text{\textquotedblleft}car" \}, C = \{ True, 90, -5 \}$ *are all examples of sets.*

## Set operations

#### Operation | #### Description | #### Graphic |

**Union of **$A$** and **$B$ ($A \cup B$B)
| Contains any elements in either set $A$ or set $B$ (including elements in both) | |

**Intersection of **$A$** and**** **$B$** **($A \cap B$) | Contains any elements that are in both set $A$ and set $B$ | |

**Complement of **$A$ ($A', A^c$ **or** $\overline{A})$ | Contains only elements that are not in $A$ | |

$A$** is a subset of **$B$ ($A \subset B$) | Signifies that all elements of $A$ are also elements of $B$ | |

## Venn diagrams

Venn diagrams are illustrated on a sample space $\xi$, E or U (sometimes called the universal set). This is the space containing all the data considered, and all sets in the Venn diagram must be subsets of the sample space. If there are elements that lie within multiple sets, these elements must lie in an overlap or intersection of the sets.

##### Example 2

*Let $A = \{ 2,3,5,7 \}, B =\{2, 4,6,8,10 \}$ be sets within the sample space $\xi = \{1,2,3,4,5,6,7,8,9,10\}$. *

*Represent these sets as a Venn diagram.*

*The element $2$ lies in both set $A$ and set $B$. *

*The elements $3,5$ and $7$ lie in set $A$ only.*

*The elements $4,6,8$ and $10$ lie in set $B$ only.*

*Similarly, the elements $1$ and $9$ lie in neither set $A$ nor set $B$. *

*Hence, the Venn diagram is of the form:*