Template:3-ary Boolean functions; BEC; nesting analysis; overview

See also: v:Studies of Euler diagrams/splits

Boolean functions can be represented as Euler diagrams. Some circles in them intersect, and some don't.
The Euler diagram on the right shows a bundle of two intersecting circles in the middle. The circle around it is a bundle on its own.

These images show an algorithmic way to find out which arguments are bundled together, and which are separate.
They can be described in terms of sets, but the concept split is more useful.
That term is used to express, that of the universe is split in two halves in a particular way.
(But without the choice which half is to be considered the set, and which the complement.)

There are three ways how two different splits can relate to each other, and three more where they are equal.
These cases can mostly be distinguished by their number of non-empty intersections. (But there are two different cases with 2.)
In these images that are the red/green overlaps with at least one black dot. Their Venn diagrams are marked with a black ring.

context
The Euler diagrams used to represent the respective equivalence class are shown in this overview.	Possible relationships between two different splits These are the cases 4, 3 and 2u. (see file description for details)
	A next step is to decompose the Boolean function into bundles like these.

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. Examples:
0, 0, 0	1, 2u, 2u	2e, 2u, 2u	3, 4, 4

Files for each BEC ordered by pattern:

0	1
BEC 0	BEC 1

2	3	4	5
BEC 2	BEC 3	BEC 4	BEC 5

15, 18, 19
BEC 15	BEC 18	BEC 19

7, 13, 16, 20, 21
BEC 7	BEC 13	BEC 16	BEC 20	BEC 21

14, 17
BEC 14	BEC 17

6, 11		9	12
BEC 6	BEC 11	BEC 9	BEC 12

8	10
BEC 8	BEC 10

two versions of BEC 9
Each of the 22 equivalence classes above is represented by the function shown in the overview file. That is essentially an arbitrary choice, although some choices are more intuitive than others. Below on the right is a different function from BEC 9.
1001 0001	0110 0100