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File:Intersections of two sets and their complements.svg

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Summary

Description

This graphic takes a step back from the concept set, and looks at a slightly more general concept henceforth called split.
A split is only the fact that the universe is split in two halves in a particular way — but without the choice which of the two sides is considered the set, and which the complement.
This choice is only non-arbitrary if one of the sides is empty. In that case the other side is the universe.
Circles in set diagrams usually imply, that the inside represents the set, and the outside represents the complement. But this graphic is inside/outside agnostic.

There are three ways two different splits can relate to each other:

  • left: The borders cross, and the resulting 4 areas are non-empty.
  • middle: The borders do not cross, and the resulting 3 areas are non-empty.
    This case can be seen as disjoint sets (row 1, middle), sets in a subset relation (row 3, right) or intersecting sets whose union is the universe (row 4, left).
  • right: One of the sets is the universe, and thus contains the other one. Only 2 areas are non-empty.

All non-empty areas in a column appear as intersection of red and green in one of the rows.
In the left column there are 4, in each middle column there are 3, and in the column on the right there are 2.
So to find out how two splits are related to each other, one has to check how many of the four possible intersections are empty.


The graphic shows only the cases where the splits are different. But there are also these cases where they are the same:

  • None of the two sides is empty. In this case also 2 of the 4 intersections are non-empty. (So it could be confused with the case where one set is the universe. This file shows both cases.)
  • One of the two sides is empty — so the other one is the universe. Only 1 of the 4 intersections is non-empty.
  • Both of the two sides are empty. This is the unique case where the whole universe is empty. So 0 of the 4 intersections are non-empty.


Compare category 3-ary Boolean functions; BEC; nesting analysis.

This SVG was created with Inkscape.
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Watchduck
You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".

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Date/TimeThumbnailDimensionsUserComment
current11:23, 29 June 2020Thumbnail for version as of 11:23, 29 June 20201,800 × 985 (78 KB)Watchduckinclude another case, color borders
23:52, 28 June 2020Thumbnail for version as of 23:52, 28 June 20201,444 × 988 (63 KB)Watchduckinclude another column
22:17, 28 June 2020Thumbnail for version as of 22:17, 28 June 20201,093 × 988 (46 KB)Watchduck{{Information |description =These (Venn and) Euler diagrams show, that the distinction between {{w|disjoint sets|}} and sets in a {{w|subset}} relation depends on how the sets are labelled. There is an actual difference between the column on the left and the two columns on the right. But the difference between the two columns on the right is only, whether the red set or its complement is considered ''the set''. |date = |source ={{own}} |author ={{Watchduck}} }} [[Cat...

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