Venn diagram

The Diagram S of Venn and the diagrams of Euler are diagrammatic representations of units, logical or mathematical relations.

Venn diagrams, Euler and Johnston

Let us consider C= {C₁,…, C_n}, a whole of simple curves closed of the plan. C is a Venn diagram , or a N-diagram of Venn, if all the intersections of an interior region (or external) delimited by a curve, with an interior region (or external) delimited by another curve are not vacuums and related, and so moreover it has there only one finished number of such intersections.

That is to say C= {C₁,…, C_n}, a whole of simple curves closed of the plan. C is a diagram of Euler , if all the intersections of an interior region (or external) delimited by a curve, with an interior region (or external) delimited by another curve are related, and so moreover there is only one finished number of such intersections.

The difference between a diagram of Euler and a Venn diagram lies only in the possibility, for intersections of areas delimited by the curves, to be empty. A Venn diagram is thus a particular case of diagram of Euler.

The Venn diagrams, Johnston, and Euler can seemingly seem identical. Their differences appear in their scopes of application according to the overall type which must be partitionné.

The diagrams of Johnston are adapted to the representation of the values of truth of the logical proposals, while the diagrams of Euler are used to illustrate specific whole of objects and the Venn diagrams are generally used to highlight possible relations.

In analysis of the Syllogisme S, the areas of the Venn diagrams are colored when they do not contain an element and in this way they make it possible to represent all the systems of units, which is more difficult with diagrams of Euler taking into account the fact that the areas not containing an element are not represented.

In fact, Euler tried to show the relationship between specific units, while Venn wanted to represent all possible partitionings of the units.

These concepts of diagram of Euler and Venn, “were not unified” and were allotted to Euler, probably because Euler introduced its diagrams 100 years earlier, and was already the author of many work.

Charts of Venn diagrams

The difficulty often consists in finding a chart pleasant of a Venn diagram. Let us point out that a Venn diagram to N units must share the plan in 2ⁿ zones, i.e. 4 zones for 2 units, 8 per three units, 16 per four…

As much it is simple to represent Venn diagram with two units (two closed curves) or even with three (three circles of radius R whose centers form an equilateral triangle on side R), as much the situation becomes complicated for the representation of a Venn diagram to four unit, even 5. Venn even forever be satisfied for him with its representation of 5 units. It is necessary to wait nearly one century before the geneticist and statistician A.W.F. Edwards does not propose of it an elegant representation in its book Cogwheels off the Mind

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