Studies of Euler diagrams/bloated

dummy

blighted

Set partitions extended with a complement relation

A Boolean function is bloated, if some of its sets are equal or complementary to each other.
(In terms of splits it means that some splits are equal.)

It is not useful to complicate an Euler diagram with that kind of information.

Instead it makes sense to separate the information, and to make an Euler diagram for the bloatless part.

The bloat part can be expressed as an extended set partition.
It has the usual blocks of equivalent elements, and the additional feature, that two blocks can be complements.

TL;DR: A bloated Boolean function has a high arity, but the gist of it can be expressed by one with a lower arity.
A Boolean function about smokers, non-smokers and wine drinkers is bloated, and should be reduced to one about smokers and wine-drinkers.
The fact that smokers and non-smokers are complementary sets does not deserve to be represented in an Euler diagram.

tokosi   (3/5)

This Boolean function has 5 arguments, but only 3 of them are independent.
Only the white areas in the Edwards–Venn diagram are true. Only the white and gray areas are interesting.

The orange and greenish areas describe the bloat:
The orange area means, that D and E are equal. The greenish area means, that B is their complement.

This is the bloat in short notation:   $B~{\Big \|}~D=E$      (The sign $\|$ is used to express, that the values on both sides are complements.)

bloated
bloatless	bloat

B was chosen to represent the redundant arguments in the Euler diagram. (Here the bloat is also shown as an Euler diagram, but generally this is not useful.)

more 5-ary examples

As in the example above, only the white areas are true.

In all these four functions A and C are complements, which is indicated by the greenish area.     $A~{\Big \|}~C$

In the last function A is also equal to E, which is indicated by the additional brown area.     $A=E~{\Big \|}~C$

To make a bloatless diagram, one has to choose which side of the complement pair should represent it. Here both variants are shown for each function.

rudafi   (4/5)

rigeko   (4/5)

megomi   (4/5)

demole   (3/5)

other examples

futare   (5/8)
${\Bigl (}A=C~{\Big \|}~D{\Bigr )}~~~{\Bigl (}B~{\Big \|}~E{\Bigr )}$
bloated	bloatless       E instead of B	topological shape without any preference	bloatless       B instead of E

geteso   (6/8)
${\Bigl (}A~{\Big \|}~B{\Bigr )}~~~{\Bigl (}G=H{\Bigr )}$
bloated	bloatless

bareto   (2/4)
This function is $(A\cap D=\varnothing )\land (B^{\complement }=C)$.   The bloatless part is $(A\cap D=\varnothing )$, and the bloat is $(B^{\complement }=C)$.     $B~{\Big \|}~C$.
bloated   (4-ary)	bloatless inflated   (2-ary inflated to 3-ary)	bloatless   (2-ary)	bloat   (2-ary)
			bloat bloatless (0-ary inflated to 1-ary)

manila   (7/8)
$B~{\Big \|}~C$
bloated	bloatless

There are also some bloated functions among the blotted examples.