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 non-empty, a split is the same as a partition into two blocks.

two splits

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 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 three cases where they are the same

all six cases

Cases where the two splits are equal: 0: Special case where the universe is empty. 1: One side is empty, so the other one is the universe. 2e Both sides are non-empty.

Cases where they are different: 2u: One of the sets is the universe, and thus contains the other one. 3: The borders do not cross, and the resulting 3 areas are non-empty. 4: The borders cross, and the resulting 4 areas are non-empty.

These four Boolean functions show examples of each case.

0, 0, 0	1, 2u, 2u	2e, 2u, 2u	3, 4, 4

examples for 3-ary Boolean functions

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 non-empty.

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
2e, 3, 3 1001 0001	2e, 3, 3 0110 0100

hypersplits

one of two 3-splits of medusa, where 3 crossing planes form 8 octants

The concept can be extended to higher dimensions. A hypersplit of dimension n (or n-split for short) partitions space into 2ⁿ orthants.
E.g. a 2-split has four quadrants, and a 3-split has 8 octants. Each orthant needs to contain at least one spot - but it can be a gapspot.

• List for medusa (two 3-splits)
• List for farofe (one 3-split)
• List for rudege (three 3-splits, one with a single gapspot as an octant)