ThatsMaths

Do you recall coming across those diagrams with overlapping circles that were popularised in the ‘sixties’, in conjunction with the “New Maths”. They were originally introduced around 1880 by John Venn, and now bear his name.

Left: Stained glass window at Gonville & Caius College, Cambridge showing a Venn diagram. Right: John Venn (1834-1923) with signature [images Wikimedia Commons].

Venn was a logician and philosopher, born in Hull, Yorkshire in 1834. He studied at Cambridge University, graduating in 1857 as sixth Wrangler in the Mathematical Tripos, that is, sixth best in mathematics in the entire university that year. He was ordained five years later, having come from a family with long traditions as churchmen. He was the eighth generation of the family to have a university education. Venn introduced the diagrams in his book Symbolic Logic, published in 1881.

The idea of a set is amongst the most fundamental concepts in mathematics. A set is any well-defined collection of distinct objects. These objects are called the members or elements of the set. They may be finite or infinite in number. Set theory was founded by the German mathematician Georg Cantor, who discovered many remarkable and counter-intuitive properties of infinite sets.

Venn Diagrams

Venn diagrams are very valuable for illustrating elementary properties of sets. They usually comprise a small number of overlapping circles; the interior of a circle represents a collection of numbers or objects or perhaps some more abstract set.

We often draw a rectangle to represent the “universe”, the set of all objects under current consideration. For example, suppose we consider all animals as the universe. The rectangle below represents this universe, and the two circles indicate subsets containing particular groups of animals. The aggregate of all the elements of the two sets is called their union.

Union of sets

The elements that are in both sets make up the intersection. If one set contains all two-legged animals and the other has all flying animals, then bears, birds and bees are in the union, but only birds are in the intersection.

Intersection of sets.

In the diagram below, the elements of our universe are all the people from Connacht. We see three subsets shown by circles: red-heads, singers and left-handers, all from Connacht. Clearly, these sets overlap and, indeed, there are some copper-topped, crooning cithogues(*) in Connacht, located in the central shaded region where the three circles overlap.

The three overlapping circles have attained an iconic status, and are used in a huge range of contexts. It is possible to devise Venn diagrams with four or more sets, but they are not so popular, as the regions can no longer all be represented by circles if all possible combinations of membership are allowed for, and the essential simplicity of the three-set diagram is lost.

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(*) A cithogue (ciotóg in Dinneen’s Foclóir)  is a left-handed person, although Google-Translate appears not to know this!

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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/