Studies of Euler diagrams/splits
Given a universe of elements, one can say, that a set does two things:
 It splits the universe in two halves,
 and it labels one half as inside (the set) and the other one as outside (the complement).
It can be useful to use a weaker construct, that does only the first. This shall be called a split.
Where both sides are nonempty, a split is the same as a partition into two blocks.
two splits
[edit  edit source]The following image shows the three ways two different splits can relate to each other:
 left: The borders cross, and the resulting 4 areas are nonempty.
 middle: The borders do not cross, and the resulting 3 areas are nonempty.
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 nonempty.
All nonempty 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 three cases where they are the same
all six cases
[edit  edit source]
Cases where the two splits are equal:

Cases where they are different:

These four Boolean functions show examples of each case.
0, 0, 0  1, 2u, 2u  2e, 2u, 2u  3, 4, 4 
examples for 3ary Boolean functions
[edit  edit source]An overview of all 22 equivalence classes can be found in the category on Commons.
In the small Venn and Euler diagrams, gray cells are empty. In the big files, the black dot represents an element, i.e. these cells are nonempty.
BEC 8  

2e, 2u, 2u 
0000 1001 
BEC 14  

2e, 2e, 2e 
1000 0001 
BEC 17  

1, 1, 1 
0000 0001 
BEC 15  

3, 3, 3 
1101 0001 
BEC 9  



hypersplits
[edit  edit source]The concept can be extended to higher dimensions. A hypersplit of dimension n (or nsplit for short) partitions space into 2^{n} orthants.
E.g. a 2split has four quadrants, and a 3split has 8 octants. Each orthant needs to contain at least one spot  but it can be a gapspot.
 List for medusa (two 3splits)
 List for farofe (one 3split)
 List for rudege (three 3splits, one with a single gapspot as an octant)