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# Venn diagram

A Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common. Usually, Venn diagrams are used to depict set intersections (denoted by an upside-down letter U). This type of diagram is used in scientific and engineering presentations, in theoretical mathematics, in computer applications, and in statistics.

The drawing is an example of a Venn diagram that shows the relationship among three overlapping sets X, Y, and Z. The intersection relation is defined as the equivalent of the logic AND. An element is a member of the intersection of two sets if and only if that element is a member of both sets. Venn diagrams are generally drawn within a large rectangle that denotes the universe, the set of all elements under consideration.

In this example, points that belong to none of the sets X, Y, or Z are gray. Points belonging only to set X are cyan in color; points belonging only to set Y are magenta; points belonging only to set Z are yellow. Points belonging to X and Y but not to Z are blue; points belonging to Y and Z but not to X are red; points belonging to X and Z but not to Y are green. Points contained in all three sets are black.

Here is a practical example of how a Venn diagram can illustrate a situation. Let the universe be the set of all computers in the world. Let X represent the set of all notebook computers in the world. Let Y represent the set of all computers in the world that are connected to the Internet. Let Z represent the set of all computers in the world that have anti-virus software installed. If you have a notebook computer and surf the Net, but you are not worried about viruses, your computer is probably represented by a point in the blue region. If you get concerned about computer viruses and install an anti-virus program, the point representing your computer will move into the black area.

This was last updated in April 2005

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