Venn diagrams are used to represent a categorical statement visually. They are often used to simplify (or extrapolate from) a logical argument.

In its simplest form, a Venn diagram is composed of a single circle. However, this usually has little meaning - more often there are two intersecting circles.

Each circle represents a category (or class) of object. The space that is contained within both of the circles represents objects that belong to both classes.

Modern logic uses several conventions to clarify things:

1. A greyed-out portion of either of the circles means that that class is empty.

2. An asterisk (or other marking used for the purpose) means that the class has at least one member.

It is possible to have more than two circles. However, any more than three quickly becomes confusing. (and more difficult to draw)

Interestingly, Venn diagrams are completely inadequate for their purpose: it's bad enough that you can't understand anything with a Venn diagram of 4 circles; what's worse is that you only get 14 regions, rather than the 16=24 you'd need!

This is the plane division by circles problem (see MathWorld with that name, which also gives sequence A014206 of Sloane's On-Line Encyclopedia of Integer Sequences). n≥1 circles divide the plane into at most n2-n+2 regions (see plane division by circles for details). This is well short of the 2n "regions" into which n sets can divide their domain. Still, the first 3 values are indeed 2,4,8, which leads to the common misconception that the sequence "correctly" continues 16,32,...

Thinking about it, it seems plausible that you'll have difficulty dividing the plane into 2n regions using only simple paths...

Venn diagrams might be neat for drawing on the blackboard, but only if you've got at most 3 sets.

Venn diagrams are pictorial representations of sets that can be used to compare and contrast characteristics of different groups. When teaching children, manipulatives are often used before drawing the diagrams; attribute blocks (which come in different sizes, shapes, colors, and thicknesses) can be sorted and placed in overlapping loops of string. Venn diagrams look (and work) something like this:

          Set A                 Set B
       _____________        __________________
     / Letters which \    / Letters containing \
    / contain curves  \  /   straight lines     \
   /                   \/                        \
  /    S               /\    T                    \
 |            C       /  \                         | 
 |                   | B  |         H              |
 |                   | P  |                E       |
  \      O            \Q /      V                 /
   \                   \/            L           /
    \                  /\    K                  /
     \ ______________ /  \ ___________________ /


(Except, you know, curved, and sometimes with more than two circles)

As you can tell from this marvelous ASCII demonstration, the intersection of Set A {S, C, O, B, P, Q} and Set B {T, B, E, P, H, Q, V, L, K} is the subset that contains letters with characteristics of both sets, namely B, P, and Q.

Venn diagrams are traditionally used in mathematics (see set theory), but these and other graphic organizers have found their way into elementary school science, social studies, and reading classrooms as a way to organize information (similar to brainstorming) before writing.

Venn diagrams for classroom use: More techical explaination and cool graphics:

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