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Venn diagram

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A graphic representation of formulas of mathematical logic, mainly formulas of the propositional calculus. A Venn diagram of variables of classical propositional logic is a selection of closed contours (with homeomorphic circumferences) which subdivides the plane into domains, some of which (e.g. , ) are marked. Each marked domain , , is put into correspondence with the formula where , , is if lies within the contour and is otherwise. The formula corresponding to the diagram as a whole is . Thus, the Venn diagram in the figure corresponds to the formula

If there are no marked domains (), the diagram corresponds to an identically-false formula, e.g. . In propositional logic, Venn diagrams are used to solve decision problems, the problem of deducing all possible pairwise non-equivalent logical consequences from given premises, etc. Propositional logic may be constructed as operations over Venn diagrams brought into correspondence with logical operations.

Figure: v096550a

The apparatus of diagrams was proposed by J. Venn [1] to solve problems in the logic of classes. The method was then extended to the classical many-place predicate calculus. Venn diagrams are used in applications of mathematical logic and theory of automata, in particular in solving the problems of neural nets.

References

[1] J. Venn, "Symbolic logic" , London (1894)
[2] A.S. Kuzichev, "Venn diagrams" , Moscow (1968) (In Russian)


Comments

The idea of Venn diagrams goes back to L. Euler and they are sometimes also called Euler diagrams.

References

[a1] P. Suppes, "Introduction to logic" , v. Nostrand (1957) pp. §9.8
[a2] G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan (1953) pp. 336ff
[a3] B. Rosser, "Logic for mathematicians" , McGraw-Hill (1953) pp. 227–228; 237ff
How to Cite This Entry:
Venn diagram. A.S. Kuzichev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Venn_diagram&oldid=12932
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098