# Venn Diagram

## Venn Diagram

Representation of one or more sets by simple closed lines in which the elements are represented by dots.

The Venn diagram, like Carroll diagrams, is a graphic diagram used to represent logical relationships between sets and the elements in these sets.

### 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 2n disjoint regions in a Venn diagram.

### Example

Here is a Venn diagram representing the sets E = {abcde, fg}, F = {cdehijk} and G = {defg, k} in the universal set U = {a, b, c, d, e, f, g, h, i, j, kmn}.

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.