Studies of Euler diagrams/examples

dummy

miniri

Euler diagram (with separated cell 0)

Euler diagram (cylindric)

graph

This is a filtrate of barita. See here.

medusa

A good Euler diagram of this Boolean function needs 3 dimensions.

Euler diagram (with separated cell 0)

Euler diagram (cylindric)

graph

Euler diagram (3D)

Venn diagram of the complement (3D)
spots 11, 13, 15	$A\cap D$ spots 9, 11, 13, 15 $A\cap \neg B\cap \neg C\cap D$ spot 9	four spheres

barogi

This representation is inflated. The Boolean function is sufficiently represented by the red-yellow Euler diagram (A, D).
The green-blue Venn diagram (B, C) does not add information. But the corresponding circles e.g. in barita are relevant. So are those in the gap variants vidita and vanatu.

Euler diagram

graph

planes (3D dual to the graph)

bareto

This is like barogi shown above, but with the additional information, that B and C are complements:     $(A\cap D=\varnothing )\land (B^{c}=C)$.
This is a 4-ary Boolean function, whose bloatless part is the 2-ary barogi.   (It is a special case, that the arguments of the bloatless and bloat part are disjoint.)

bloated   (4-ary)	bloatless inflated   (2-ary inflated to 3-ary)	bloatless   (2-ary)

basori

This representation is also inflated. Only the red-yellow-brown Euler diagram (A, D, E) matters.
The green-blue Venn diagram (B, C) does not add information. But the corresponding circles in basiga are relevant.

Euler diagram

graph

barita

The green-blue (B, C) and brown-magenta (E, F) Venn diagrams would be separate bundles, if they were not trisected by the red-yellow Euler diagram (A, D).
In the graph this is a multiplication, and in the formula it is a conjunction. Compare the filtrates.

Euler diagram and graph

layers of 3D Euler diagram
The 3-dimensional Euler diagram has 3*7 = 21 cells, 3*8 + 2*7 = 38 faces, 22 edges and 4 vertices. The edges and vertices are two times the arrangement seen here. Compare bar. One could get the impression, that brown and green are parallel, while brown and blue are not. But each cross could be flipped upside-down, without changing the topology.

formula tree
((!b and !c) or (!e and !f)) and !(a and d)

basiga

Without the small bundle (F, G, H) the surrounding one would be just the red-yellow-brown Euler diagram (A, D, E), as in basori.
But as the small bundle is only in B, and not in C, the green and blue circles are also needed.

If the inner bundle were on its own, it would fall apart into the circles F and H. The circle G would just bisect that Boolean function, adding no information. But in the nested bundle, the circle G is relevant. Removing it would mean, that G implicitly bisects the whole Boolean function (including A...E). But actually it bisects only one cell of the surrounding bundle, namely the one where only B and D intersect.

Compare the gap variants and filtrates.

Euler diagram and graph

formula tree

putuki

Euler diagram (matrix)        graph
Compare piferi, where spot 0 is a gap.

Euler diagram (cylinder)        formula tree
This Euler diagram is similar to that of dukeli below.	!(a and c) and !(b and d)

Venn diagram of the complement (3D)
The complement of putuki is $(A\cap C)\cup (B\cap D)$. The Venn diagram is easily recognized as a union of two lenses.
spots [5, 7, 10, 11, 13, 14, 15]	$A\cap C$ [5, 7, 13, 15] $B\cap D$ [10, 11, 14, 15]	four spheres

dukeli

This is like putuki without spot 9.   For the NP equivalence class see dukeli NP.   For a conversion example see dukeli and netuno.

Venn diagram of the complement (3D)
The complement of dukeli is $(A\cap C)\cup (A\cap D)\cup (B\cap D)$. The Venn diagram is easily recognized as a union of three lenses.
spots [5, 7, 9, 10, 11, 13, 14, 15]	$A\cap C$ [5, 7, 13, 15] $A\cap D$ [9, 11, 13, 15] $B\cap D$ [10, 11, 14, 15]	four spheres