# 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  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.

How to Cite This Entry:
Venn diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Venn_diagram&oldid=12932
This article was adapted from an original article by A.S. Kuzichev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article