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A '''Venn diagram''' is a visual way of representing the [[mathematics | mathematical]] relationship between [[set]]s.

== Two Set Example ==
The following diagram is a Venn diagram for sets <math>A</math> and <math>B</math>:

<center>[[Image:Venn1.PNG]]</center>

The red [[region]] contains all the [[element]]s that are in <math>A</math> only.  The blue region contains all the elements that are in <math>B</math> only.  The black region contains all the elements in both <math>A</math> and <math>B</math> which is called the [[intersection]] of <math>A</math> and <math>B</math>, denoted <math>A\cap B</math>.  The red, black, and blue regions together represent the elements that are in <math>A</math>, <math>B</math>, or both.  This is called the [[union]] of <math>A</math> and <math>B</math>, denoted <math>A\cup B</math>.

If we consider the region bounded by the [[rectangle]] to be the [[universal set]], then the gray area is called the [[complement]] of <math>A\cup B</math> -- that is, the things which are neither in <math>A</math> nor in <math>B</math>.  

All of this information can be summarized in the following table:

{| class="wikitable" style="margin: 1em auto 1em auto; width:500px"
! Region (by color) !! Description !! Notation
|-
| Red
| elements in <math>A</math> only
| <math>A - (A\cap B)</math>
|-
| Blue 
| elements in <math>B</math> only
| <math>B - (A\cap B)</math>
|-
| Black
| elements in both <math>A</math> and <math>B</math>
| <math>(A\cap B)</math>
|-
| Gray
| elements in neither <math>A</math> nor <math>B</math>
| <math>(A\cup B)^C</math>
|-
|
|
| or <math>(A\cup B)'</math>
|-
|
|
| or <math>\overline{(A\cup B)}</math>
|-
|
|
| or <math>U - (A\cup B)</math>
|}

== Three Set Example ==
The following diagram is a Venn diagram for the sets <math>A, B</math> and <math>C</math>.

<center>[[Image:Venn2.PNG]]</center>

The following table describes the various regions in the diagram:
{| class="wikitable" style="margin: 1em auto 1em auto; width:1000px"
! Region (by color) !! Description !! Notation
|-
| Blue 
| elements in <math>A</math> only
| <math>A - (A\cap B)-(C\cap A) + (A\cap B\cap C)</math>
|-
| Yellow
| elements in <math>B</math> only
| <math>B - (A\cap B) - (B\cap C) + (A\cap B\cap C)</math>
|-
| Red
| elements in <math>C</math> only
| <math>C - (B\cap C)-(C\cap A) + (A\cap B\cap C)</math>
|-
| Green
| elements in both <math>A</math> and <math>B</math> but not <math>C</math>
| <math>(A\cap B) - (A\cap B\cap C)</math>
|-
| Orange
| elements in both <math>B</math> and <math>C</math> but not <math>A</math>
| <math>(B\cap C) - (A\cap B\cap C)</math>
|-
| Purple
| elements in <math>C</math> and <math>A</math> but not <math>B</math>
| <math>(C\cap A) - (A\cap B\cap C)</math>
|-
| Black
| elements in <math>A,B</math> and <math>C</math>
| <math>(A\cap B\cap C)</math>
|-
| Gray
| elements in neither <math>A,B</math> or <math>C</math>
| <math>(A\cup B\cup C)^C</math>
|-
|
|
| or <math>(A\cup B\cup C)'</math>
|-
|
|
| or <math>\overline{(A\cup B\cup C)}</math>
|-
|
|
| or <math>U - (A\cup B\cup C)</math>
|}

== Using Venn Diagrams ==
Venn diagrams are very useful for organizing data.  In particular, the [[Principle of Inclusion-Exclusion]] can be explained for small cases nicely using them.

== External links ==
* [http://theory.cs.uvic.ca/~cos/venn/VennEJC.html A Survey of Venn Diagrams]

== See also ==
* [[Combinatorics]]

[[Category:Combinatorics]]
[[Category:Definition]]