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# Venn diagram

Note: Currently, overlapping colors do not seem to be showing up properly in html mode. Therefore, this entry is best viewed using page images mode.

A *Venn diagram* is a visual tool used in describing how two or more sets are logically related to one another. The simplest example is a Venn diagram for two sets. Each set is represented by a planar region bounded by a circle, so that the two regions overlap. In the diagram below, set $A$ is represented by the reddish circular disc and set $B$ is represented by the bluish circular disc:

The overlapping region represents the intersection of the sets $A$ and $B$, denoted by $A\cap B$.

Typically, the two sets are subsets of some bigger set $U$, called a universe. Therefore, the corresponding Venn diagram above is shown to be sitting inside a larger rectangular region representing the universe:

Notice that the region representing the universe $U$ is partitioned into $4$ mutually exclusive regions:

Region | Set being represented | Symbol |
---|---|---|

$1$ | the intersection of $A$ and $B$ | $A\cap B$ |

$2$ | $A$ excluding $B$ | $A\cap B^{{\prime}}$ |

$3$ | $B$ excluding $A$ | $A^{{\prime}}\cap B$ |

$4$ | neither $A$ nor $B$ | $A^{{\prime}}\cap B^{{\prime}}$ |

A Venn diagram for three sets is slightly more complicated, and is illustrated below:

As indicated by the diagram, $U$ is divided into $8$ regions. The $8$ regions represent the following sets (where $A,B,C$ are sets represented by circular discs Red, Blue, Yellow, respectively):

Region | Set | Region | Set | Region | Set | Region | Set |
---|---|---|---|---|---|---|---|

$1$ | $A\cap B^{{\prime}}\cap C^{{\prime}}$ | $2$ | $A\cap B\cap C^{{\prime}}$ | $3$ | $A^{{\prime}}\cap B\cap C^{{\prime}}$ | $4$ | $A\cap B^{{\prime}}\cap C$ |

$5$ | $A\cap B\cap C$ | $6$ | $A^{{\prime}}\cap B\cap C$ | $7$ | $A^{{\prime}}\cap B^{{\prime}}\cap C$ | $8$ | $A^{{\prime}}\cap B^{{\prime}}\cap C^{{\prime}}$ |

More generally, a Venn diagram may be constructed for any finite number of sets, and the regions representing the sets need not be circular discs, as long as each region is the interior of a simple closed curve (in the plane $\mathbb{R}^{2}$). Specifically, a Venn diagram $V$ for $n$ sets is a set of $n$ simple closed curves $C_{1},\ldots,C_{n}$ in $\mathbb{R}^{2}$ such that

$A_{1}(k_{1})\cap\cdots\cap A_{n}(k_{n})\neq\varnothing$ |

for every combination of $(k_{1},\ldots,k_{n})$, where each $k_{i}\in\{0,1\}$, and that $A_{i}(0)$ is the region $A_{i}$ bounded by $C_{i}$, and $A_{i}(1)$ is $A_{i}^{{\prime}}$, the complement of $A_{i}$ in $\mathbb{R}^{2}$. In a nutshell, this says that the curves $C_{1},\ldots,C_{n}$ partition $\mathbb{R}^{2}$ into $2^{n}$ non-empty regions. This is in fact possible for every $n<\infty$. Furthermore, if we treat a straight line as a circle whose center is located at $\infty$, the possibility of constructing a Venn diagram for $n$ sets is the same as saying the possibility of constructing $n$ straight lines in $\mathbb{R}^{2}$ such that the lines partition the plane into $2^{n}$ regions.

Remark. Note that given a Venn diagram for $n$ sets, no assumptions are made concerning these $n$ sets, although it is generally assumed that the logical structure of these sets correspond to the logical structure of the regions bounded by the closed curves in the Venn diagram. For example, it is possible that $A\cap B=\varnothing$, or that $A\subseteq B$. A Venn diagram for both examples would look the same, although $A\cap B$ represents $\varnothing$ in the former case, and $A$ in the latter one.

## Mathematics Subject Classification

00A05*no label found*00A06

*no label found*03E99

*no label found*

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## Comments

## venn diagrams

Hi, Chi,

I downloaded a couple of XY-pic and PSTricks docs but haven't gotten around to reading them yet, so maybe I'll try this the lazy way.

I've been looking at the code for your 3-circle venn diagram.

Is there an easy way to fill individual cells with shading?

For example, I need to convert the venn diagrams in MinimalNegationOperator from ASCII to better form.

TIA,

Jon Awbrey

## Re: venn diagrams

Jon,

If you are interested in overlaying colors on top of each other, the best I could come up is in the Venn Diagram entry.

If you are interested in coloring individual cells, for example, coloring the intersection of two regions, you may also want to look at the entry on hyperbolic angles:

http://planetmath.org/encyclopedia/HyperbolicAngle.html

By the way, I think there is a need on PM to have a compilation of illustrated examples of how to create pictures and diagrams using pstricks and xypic. Some progress has been made regarding this front. But it is still far from being perfect.

Please look under the collaborations section, and in particular

http://planetmath.org/?op=getobj&from=collab&id=53

and

http://planetmath.org/?op=getobj&from=collab&id=138