Venn diagrams

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Summary

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,car"},C={True,90,5}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 AA​ and BB (ABA \cup BB)

Contains any elements in either set AA or set BB (including elements in both)


Maths; Probability; KS3 Year 7; Venn diagrams

Intersection of AA and BB (ABA \cap B)

Contains any elements that are in both set AA and set BB

Maths; Probability; KS3 Year 7; Venn diagrams

Complement of AA (A,AcA', A^c or A)\overline{A})

Contains only elements that are not in AA​​

Maths; Probability; KS3 Year 7; Venn diagrams

​​AA is a subset of BB (ABA \subset B)

Signifies that all elements of AA are also elements of BB​​

Maths; Probability; KS3 Year 7; Venn diagrams




​​Venn diagrams

Venn diagrams are illustrated on a sample space ξ\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}A = \{ 2,3,5,7 \}, B =\{2, 4,6,8,10 \} be sets within the sample space ξ={1,2,3,4,5,6,7,8,9,10}\xi = \{1,2,3,4,5,6,7,8,9,10\}.  

Represent these sets as a Venn diagram.


The element 22​ lies in both set AA and set BB.  


The elements 3,53,5​ and 77 lie in set AA only.


The elements 4,6,84,6,8 and 1010 lie in set BB only.


Similarly, the elements 11​ and 99​ lie in neither set AA nor set BB


Hence, the Venn diagram is of the form:


Maths; Probability; KS3 Year 7; Venn diagrams


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Exercises

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FAQs - Frequently Asked Questions

What is a set?

Why do we use a Venn diagram?

What does {} mean?

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