Extensions To Higher Numbers of Sets
Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that is completely symmetric and can be visually represented. The 16 intersections correspond to the vertices of a tesseract (or the cells of a 16-cell respectively).
For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures…elegant in themselves," that represented higher numbers of sets, and he devised a four-set diagram using ellipses (see below). He also gave a construction for Venn diagrams for any number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.
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Venn's construction for 4 sets
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Venn's construction for 5 sets
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Venn's construction for 6 sets
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Venn's four-set diagram using ellipses
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This symmetrical or Euler diagram is not a Venn diagram for four sets as it has only 13 regions (excluding the outside); there is no region where only the yellow and blue, or only the pink and green circles meet.
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Five-set Venn diagram using congruent ellipses in a radially symmetrical arrangement devised by Branko Grünbaum. Labels have been simplified for greater readability; for example, A denotes A ∩ Bc ∩ Cc ∩ Dc ∩ Ec (or A ∩ ~B ∩ ~C ∩ ~D ∩ ~E), while BCE denotes Ac ∩ B ∩ C ∩ Dc ∩ E (or ~A ∩ B ∩ C ∩ ~D ∩ E).
Read more about this topic: Venn Diagram
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