### Properties

A Venn diagram is made up of curved closed lines inside of which are gathered the elements of the sets that they represent, so that:

- each element is identified by a capital letter that represents it;
- each element of the universal set is represented only once and is identified by its name (letter, number, etc.);
- each element in the universal set belongs to only one region of the diagram;
- an empty region of the diagram is cross-hatched;
- the intersections of the curved lines in the diagram are placed so that an attribute is represented by only one region; these regions are disjointed from one another.

The result of these conditions is that:

- the representation of the subset E of a universal set U produces two regions corresponding to the attributes “belongs to” and “does not belong to” set E;
- the representation of two subsets E and F of a universal set produces four regions corresponding to the following subsets: E ∩ F, E \ F, F \ E and (E ∪ F)’;
- the representation of three subsets E, F and G of a universal set U produces 8 regions corresponding to each of the eight disjoint subsets;
- the representation of
*n*subsets of a given universal set U produces 2_{n}disjoint regions in a Venn diagram.

### Example

Here is a Venn diagram representing the sets E = {*a*, *b*, *c*, *d*, *e, **f*, *g*}, F = {*c*, *d*, *e*, *h*, *i*, *j*, *k*} and G = {*d*, *e*, *f*, *g*, *k*} in the universal set U = {*a*, *b*, *c*, *d*, *e*, *f*, *g*, *h*, *i*, *j*, *k*, *m*, *n*}.

This diagram includes 8 disjoint regions.

### Historical Note

John Venn (1834-1923) was a British mathematician and logician who, in 1881, introduced the representation in the form of regions – sometimes called *ovals* – of sets of objects. In doing so, he revisited a similar mode of representation proposed by Euler before him by making improvements to it including identifying empty regions, using regions that are not necessarily circular, etc.