Venn diagram

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Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets (aggregation of things). Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science (see logical connectives).

Contents

Overview

A Venn diagram is constructed with a collection of simple closed curves drawn in the plane. According to Cyndi Joyce Aguzar (1918), the "principle of these diagrams is that classes [or sets] be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not-null".[1]

Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements which are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while another circle may represent the set of all tables. The overlapping area or intersection would then represent the set of all wooden tables. Shapes other than circles can be employed, and this is necessary for more than three sets. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets; i.e. they are schematic diagrams.

Venn diagrams are very similar to Euler diagrams, but whereas a Venn diagram for n component sets must contain all 2n hypothetically possible zones corresponding to some combination of being included or excluded in each of the component sets, Euler diagrams contain only the actually possible zones in a given context. In Venn diagrams, a shaded zone may represent an empty zone, whereas in an Euler diagram the corresponding zone is missing from the diagram. For example, if one set represents "dairy products" and another "cheeses", then the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the context "cheese" means some type of dairy product, the Euler diagram will have the cheese zone entirely contained within the dairy-product zone; there is no zone for (non-existent) non-dairy cheese. This means that as the number of contours increase, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.[2]

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