[Math Lair] Venn Diagrams

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Venn diagrams are a way of visualizing information about one or more sets, their elements, and the logical relationships between them. They are named for John Venn, who used them in 1876 in a paper on boolean algebra. They are helpful when investigating problems relating to boolean algebra, categorical syllogisms and other problems in logic and set theory.

A Venn diagram usually contains circular or elliptical areas that represent sets. For example, the relationship between the set of prime numbers (denoted here by P), the set of triangular numbers (T) and the set of even numbers (E) could be represented as follows:

[Sample Venn diagram]

Some numbers, such as 9, 25, and 27, are neither prime, triangular, nor even. These have been placed outside of all three circles. Other numbers, such as 4, 8, and 14, are even but not triangular or prime. These have been placed inside the E circle but outside the other two circles. 2 is prime and even, but not triangular, so it has been placed inside the E and P circles but not the T circle. There are no numbers that are prime, even, and triangular, so there is nothing in the middle of the diagram, where all three sets intersect (we could, if desired, shade the intersection of all three sets to show that it is empty).

One difficulty with Venn diagrams is that, when working with four or more sets, Venn diagrams can be difficult to draw. Venn was only able to create Venn diagrams with no more than four sets. It took over a century to show that a symmetric Venn diagram can be made from five ellipses.

Sources used (see bibliography page for titles corresponding to numbers): 51.