Venn diagram

Venn diagrams, Euler diagrams (pronounced "oiler") and Johnston diagrams are similar-looking illustrations of set, mathematical or logical relationships.

Example

Sets A and B

The Venn diagram above can be interpreted as "the relationships of set A and set B which may have some (but not all) elements in common".

Sets A, B and C

The Euler diagram above can be interpreted as "set A is a proper subset of set B, but set C has no elements in common with set B.

Or, as a syllogism

All Vs are Ts
All Ks are Vs
Therefore All Ks are Ts.

Venn, Johnston, and Euler diagrams may be identical in appearance. Any distinction is in their domains of application, that is in the type universal set that is being divided up. Johnston's diagrams are specifically applied to truth values of propositional logic, whereas Euler's illustrate specific sets of "objects" and Venn's concept is more generally applied to possible relationships. It is likely that the Venn and Euler versions have not been "merged" because Euler's version came 100 years earlier, and Euler has credit for enough accomplishment already, whereas John Venn has nothing left to his name but the diagram.

The difference between Euler and Venn may be no more than that Euler's try to show relationships between specific sets, whereas Venn's try to include all possible combinations. With that in mind:

There was some struggle as to how to generalise to many sets. Venn got as far as four sets by using ellipses:

image:venn-four.png

but was never satisfied with his five-set solutions. It would be more than a century before a means satisfying Venn's somewhat informal criteria of ‘symmetrical figures…elegant in themselves’ was found. In the process of designing a stained-glass window in memoriam to Venn, Anthony Edwards came up with ‘cogwheels’:

three sets: image:Edwards-Venn-three.png

four sets: image:Edwards-Venn-four.png

five sets: image:Edwards-Venn-five.png

six sets: image:Edwards-Venn-six.png

Ref: Ian Stewart Another Fine Math You've Got Me into 1992 ch4

External links

LogicTutorial.com - interactive Johnston diagram
Lewis Carroll's Logic Game - Venn vs. Euler