The Basics of Venn Diagrams

An Important Tool in Set Theory and Beyond

Aug 15, 2008 Isaac M. McPhee

Created by British Philosopher John Venn in 1881, Venn diagrams have made their way into almost every facet of set-based thought, well beyond mathematics.

Most people can easily recognize Venn diagrams, even if not aware of their name. They have become rather common place in society since the time of their introduction well over a century ago, and have proven endlessly useful in providing a visual representation regarding the relationships between various groups of things.

The Basics: How Venn Diagrams Work

Simply put, Venn diagrams most often consist of nothing more complicated than circles, usually just two or three (though there have been many who have experimented with Venn diagrams of a higher order than this, with some absolutely outrageous results).

Each of the circles in a Venn diagram represents a specific set. In mathematics, perhaps one circle is the set of all integers between 50 and 100, while a second is the set of all integers between 90 and 150. As one can plainly see, there are several numbers (90-100) which fall into both sets. Because of this, a Venn diagram would show these two sets overlapping, and this overlap would represent these numbers which adhere to the limitations posed by both sets at once.

This overlap, in set theory, is known as the "intersection" between the two sets.

Now, it wouldn't be difficult to add a third set to this diagram, perhaps it is the set of all multiples of ten between 1 and 200. Now things become a bit more complicated, but still manageable. This third set clearly overlaps both of the first sets, while possessing certain members which are not part of any of the others. The result is a fairly typical Venn diagram of order 3 (as can be seen in the image accompanying this article).

Uses of Venn Diagrams

As mentioned before, Venn diagrams are certainly not limited to mathematical applications relating to groups of numbers. Rather, Venn diagrams have found use in almost every set-based system.

They are used very frequently in toxicological applications, being beneficial in showing how various plants and animals may be grouped into various categories based on similar and dissimilar features, as well as in marketing, where they have become a staple of PowerPoint presentations in boardrooms everywhere ("Circle A represents potential customers over the age of 40, while Circle B represents potential customers under the age of 40, and the intersection between these groups represents the actual number of people who will buy our product...")

As mentioned before, while Venn diagrams are most commonly used to show the relationship between just two or three groups, some very clever thinkers have found interesting ways to use these diagrams to show the relationship between even up to 7 sets! Needless to say, it is rather impossible to do this with perfect circles, and requires a certain "flexibility" to the shapes being used, which end up being "blobs" which can be stretched out and pulled and curved at will (an example of one of these is also found in the images below).

Clearly, Venn Diagrams have only just begun to show their full potential.

The copyright of the article The Basics of Venn Diagrams in Math/Chaos Theory is owned by Isaac M. McPhee. Permission to republish The Basics of Venn Diagrams in print or online must be granted by the author in writing.

A Basic 3-Part Venn Diagram

An extreme Venn Diagram