Studies of Euler diagrams/blotted

dummy

blighted

A Boolean function is blotted, if one of its sets is equal or complementary to the universe.
(In terms of splits it means that there is a one-sided split.)

This is similar to bloated functions.
The fact that some set contains everything or nothing should be duly noted,
but not be mixed with the more interesting information to be expressed in an Euler diagram.

fobope   (3/4)

This is a 4-ary Boolean function. It contains the usual kind of information about A, B and D, as well as the fact that C is empty.
The Euler diagram on the left does not make sense. It may have been drawn this way, because generally C is in B (see here),
but when C is empty, this becomes meaningless, because the empty set is a subset of every set.
In the Euler diagram on the right the fact that nothing is C is expressed by the light C on the outside of the universe.

$\mathrm {U} ~{\Big \|}~\varnothing =C$

blotted   (4-ary)

blotless   (3-ary)

lapava   (7/8)

This is an 8-ary Boolean function. It contains the usual kind of information about A...G, as well as the fact that there is nothing outside of H.
The Euler diagram on the left is more complicated than necessary, because the ellipse for H is shown separately from the box for the universe.
In the Euler diagram on the right the fact that everything is H is expressed by the light H on the inside of the universe.

$H=\mathrm {U} ~{\Big \|}~\varnothing$

blotted   (8-ary)

blotless   (7-ary)

other examples

Most of these examples are also bloated. (Only todeda and gufaro are not.)

menose   (0/3)
This function (the complement of safoko) contains no blightless information — i.e., the blightless part is the tautology. The blot is that C is empty. The remaining bloat is that A and B are complements. ${\Bigl (}\mathrm {U} ~{\Big \|}~\varnothing =C{\Bigr )}~~~{\Bigl (}A~{\Big \|}~B{\Bigr )}$

todeda   (7/8)
$H=\mathrm {U} ~{\Big \|}~\varnothing$

gufaro   (6/7)
$\mathrm {U} ~{\Big \|}~\varnothing =H$

temapi   (3/7)
$E=\mathrm {U} ~{\Big \|}~\varnothing =A=B=D$

tinefa   (5/8)
$E=\mathrm {U} ~{\Big \|}~\varnothing =D=H$

bitada   (5/8)
${\Bigl (}B=\mathrm {U} ~{\Big \|}~\varnothing =F{\Bigr )}~~~{\Bigl (}A=C{\Bigr )}$

duvola   (4/8)
${\Bigl (}B=\mathrm {U} ~{\Big \|}~\varnothing =F{\Bigr )}~~~{\Bigl (}A=C~{\Big \|}~D{\Bigr )}$