What Is a Venn Euler Diagram?
An Euler (pronounced "oiler") diagram is a picture that represents sets and their relationships. It consists of closed curves, the elements of which are represented by the interior of the curves. The relationships to be represented are overlap, containment or neither. A Venn diagram represents all possible overlap permutations between sets. Therefore, Venn diagrams are a subset of Euler diagrams, for they are the Euler diagrams that represent all potential overlaps, whether the sets have elements in common or not.
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Euler Circles
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Euler circles were used by Leonard Euler in the 18th century to represent relationships between sets. The relationships possible included overlap, containment and neither. For example, a circle inside of a circle represents containment. Socrates was a man, but not all men are Socrates. An Euler circle representing Socrates is contained in a circle representing all men.
Venn Diagrams
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Euler diagrams do not present null sets, but Venn diagrams do. For example, in a Venn diagram, two sets are represented as overlapping, whether they share elements or not. That the overlap is empty would be specified outside of the diagram. If such information must be presented graphically, Euler diagrams that leave out the empty overlap are used.
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Higher-Order Venn Diagrams
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Representing all overlap permutations for two or three sets is easy. For four, it starts getting hard. A solution for how to represent all the overlap permutations for five sets is in the adjacent diagram.
Use of Venn Diagrams
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What do Venn diagrams add? It seems, after all, that they take information away, by hiding whether an overlap is empty or not. Of the three relationships mentioned before, "containment" and "neither" aren't represented. John Venn constructed his diagrams for the purposes of logic, to formulate observations independent of whether any given overlap mix is empty or not. By representing every possibility, one can take simple logic statements such as, "the complement of the intersection of A and B is the union of the complement of A and the complement of B," and prove it with a Venn diagram, or see if it generalizes to higher orders, i.e. see if it can be extended when more sets are added in.
Venn Diagrams in Probability
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Another such use for Venn diagrams arises in probability, when sets represent events. When calculating the probability of A or B happening, and both events are possible, then their overlap is not necessarily empty. Venn diagrams help keep track that, when adding the probability of A and B, one then subtract out the overlap of A and B to avoid double-counting it. When the event count increases to three, the number of regions increases precipitously, and Venn diagrams become particularly helpful in avoiding double-counting.
Whether an overlap is empty or not, a diagram that represents all the possibilities helps with the accounting. If an overlap is known to have no probability, then it is helpful to revert to an Euler diagram and remove the overlap from the diagram.
Introductory statistical textbooks such as Freund and DeGroot identify Venn diagrams as such, but refer to Euler diagrams merely as "diagrams."
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