Euler diagram

﻿
Euler diagram

Euler diagrams or "Euler circles" are a diagrammatic means of representing sets and their relationships. They are the modern incarnation of Euler circles, which were invented by Leonhard Euler in the 18th century.

Overview

Euler diagrams usually consist of simple closed curves in the plane which are used to depict sets. The spatial relationships between the curves (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).

Euler diagrams generalise the well-known Venn diagrams which represent all possible set intersections available with the given sets.

The intersection of the interior of a collection of curves and the exterior of the rest of the curves in the diagrams is called zone. Thus, in Venn diagrams all zones must be present (given the set of curves), but in an Euler diagram some zones might be missing.

In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples on the right, the Euler diagram depicts that the sets "Animal" and "Mineral" are disjoint since the corresponding curves are disjoint, and also that the set "Four Legs" is a subset of the set of "Animal"s. The Venn diagram which uses the same categories of "Animal", "Mineral" and "Four Legs" does not encapsulate these relationships. Traditionally the "emptiness" of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent "emptiness" either by shading or by the use of a missing zone.

Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the diagram below, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets which are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.

ee also

* Johnston diagram

References

* Euler Diagrams. Brighton, UK (2004). [http://www.cs.kent.ac.uk/events/conf/2004/euler/eulerdiagrams.html What are Euler Diagrams?]
* [http://www.eulerdiagrams.com/ Visualisation with Euler diagrams project]

Wikimedia Foundation. 2010.

Look at other dictionaries:

• euler diagram — …   Useful english dictionary

• euler's diagram — noun see euler diagram * * * Logic. one of a number of graphic representations of the logical relations among classes by means of relations among circles or other geometric figures. [named after L. EULER] * * * Euler s diagram, a graphic… …   Useful english dictionary

• Diagram — Further information: Chart Sample flowchart representing the decision process to add a new article to Wikipedia. A diagram is a two dimensional geometric symbolic representation of information according to some visualization technique. Sometimes …   Wikipedia

• Euler's diagram — Logic. one of a number of graphic representations of the logical relations among classes by means of relations among circles or other geometric figures. [named after L. EULER] * * * …   Universalium

• Venn diagram — Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets (groups of things). Venn diagrams were invented around 1880 by John Venn. They are used in many fields,… …   Wikipedia

• List of topics named after Leonhard Euler — In mathematics and physics, there are a large number of topics named in honour of Leonhard Euler (pronounced Oiler ). As well, many of these topics include their own unique function, equation, formula, identity, number (single or sequence), or… …   Wikipedia

• Spider diagram — A spider diagram adds existential points to an Euler or a Venn diagram. The points indicate the existence of an attribute described by the intersection of contours in the Euler diagram. These points may be joined together forming a shape like a… …   Wikipedia

• Leonhard Euler — Infobox Scientist name = Leonhard Euler|box width = 300px |200px image width = 200px caption = Portrait by Johann Georg Brucker birth date = birth date|df=yes|1707|4|15 birth place = Basel, Switzerland death date = 18 September (O.S 7 September)… …   Wikipedia

• Johnston diagram — Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. Logically they are equivalent to truth tables; some may find them easier to understand at a glance. By overlaying one… …   Wikipedia

• Van Kampen diagram — In the mathematical area of geometric group theory, a van Kampen diagram is a planar diagram used to represent the fact that a particular word among the generators of a group given by a group presentation represents the identity element in that… …   Wikipedia