Studies of Euler diagrams/clans/tables

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Tabita is a 4-ary BF. The table is shown with a small hover table in this screenshot. It can be found in 4/block sizes 4*3.

More examples can be found on the subpages for 3-ary and 4-ary Boolean functions.

The natural subdivisions of a clan of Boolean functions are families (negating arguments) and factions (permuting arguments).

These tables show all BF of a clan with the factions as rows and the families as grouped columns.

The BF are represented by their Zhegalkin indices, and the order of the rows and columns is based on these numbers.
The chosen representative is highlighted in violet, and its name is shown in the header.
The transformations, denoted by number icons, refer to this representative. (Otherwise it has no influence on the table.)

The following table shows the clan of takate (Ж 30) and its complement gilera (Ж 31).
It has three families, each with 8 BF.
Its six factions contain 3 BF if the Venn diagrams are symmetric, or 6 BF if they are not.

takate

W C 24 36 66 2 3 24 &3_3 3 24 (0, 1, 2) (1, 0, 2) 36 (0, 2, 1) (2, 0, 1) 66 (1, 2, 0) (2, 1, 0) 3 3 25 &2_2 2 25 (0, 1, 2) (1, 0, 2) 37 (0, 2, 1) (2, 0, 1) 67 (1, 2, 0) (2, 1, 0) 3 6 26 &2_2 2 26 (0, 1, 2) (1, 0, 2) 28 (1, 0, 2) (0, 1, 2) 38 (0, 2, 1) (2, 0, 1) 52 (2, 0, 1) (0, 2, 1) 70 (1, 2, 0) (2, 1, 0) 82 (2, 1, 0) (1, 2, 0) 4 6 27 &1_1 1 27 (0, 1, 2) (1, 0, 2) 29 (1, 0, 2) (0, 1, 2) 39 (0, 2, 1) (2, 0, 1) 53 (2, 0, 1) (0, 2, 1) 71 (1, 2, 0) (2, 1, 0) 83 (2, 1, 0) (1, 2, 0) 4 3 30 &0_0 0 30 (0, 1, 2) (1, 0, 2) 54 (0, 2, 1) (2, 0, 1) 86 (1, 2, 0) (2, 1, 0) 5 3 31 &1_1 1 31 (0, 1, 2) (1, 0, 2) 55 (0, 2, 1) (2, 0, 1) 87 (1, 2, 0) (2, 1, 0)

mouseovers of Zhegalkin indices
Hovering over the numbers will show a mouseover. Its first line is the Zhegalkin index in little-endian binary (which corresponds to the algebraic normal form). The second line is the truth table of the BF. The screenshot on the right shows a mouseover in the table above. (The diagram to its right shows how the second line is derived from the first.)

The number icons denote the transformations from the representative to each particular BF.
A transformation is a signed permutation, i.e. a negator pattern paired with a permutation — both of which can be represented by a natural number.
While the permutation index numbers are unambiguous, there is some potential confusion about numbering the negator patterns.
Generally it is more useful to refer to which values are negated. (See this 8×6 matrix.)
But BF in the same faction have the same negated places in common — which are thus used as row labels in these tables.
To avoid confusion, the icons referring to negated values are round, and those referring to negated places are square.

signed permutation (~1, 2, ~0)

This signed permutation is based on the permutation (1, 2, 0) with the index number . The distribution of negators will usually be described as (1, 1, 0) , based on which values are negated. (Seen in the rows of the matrix.) But it can also be described as (1, 0, 1) , based on which places are negated. (Seen in the columns of the matrix.) python code This code uses the Python library discrete helpers. import numpy as np from discretehelpers.sig_perm import SigPerm sigperm = SigPerm(pair=[3, 4]) assert sigperm == SigPerm(sequence=[~1, 2, ~0]) assert sigperm.sequence_string() == '[~1, 2, ~0]' assert sigperm.matrix() == np.array([ [ 0, 0, -1], [-1, 0, 0], [ 0, 1, 0] ]) assert sigperm.keyneg_index == 5 # 101 is the position of minus signs in the matrix columns assert sigperm.valneg_index == 3 # 110 is the position of minus signs in the matrix rows assert sigperm.perm_index == 4

So each BF corresponds to a set of transformations.
To such a set corresponds a set of keyneg indices, a set of valneg indices and a set of perm indices.
Factions (rows) have the same set of keyneg indices, and families (grouped columns) have the same set of perm indices in common.
The sets of perm indices are cosets of The symmetric group $S_{n}$, where $n$ is the arity. (See Subgroups of S₄.)
The valneg set is shown under each Zhegalkin index. Hovering over it will show a table with the whole set of transformations.

transformations from takate (Ж 30) to tabora (Ж 38)

The representative BF of the table above is takate (Ж 30). The hover table is shown for tabora (Ж 38). Below is shown how the two transformations turn the Euler diagram of the former into (mirrored) Euler diagrams of the latter. takate tabora tabora hover table     ${\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}\\{\color {Red}\neg 0}&{\color {Blue}~2}&{\color {ForestGreen}\neg 1}\\\end{pmatrix}}$     ${\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}\\{\color {Blue}~2}&{\color {Red}\neg 0}&{\color {ForestGreen}\neg 1}\\\end{pmatrix}}$

Often the set of valneg indices of a BF contains only a single element.
Two are also common:

makoto

W C 44 74 98 3 6 44 &5_23 2 3 44 (0, 1, 2) (0, 2, 1) 56 (0, 2, 1) (0, 1, 2) 74 (1, 0, 2) (1, 2, 0) 88 (1, 2, 0) (1, 0, 2) 98 (2, 0, 1) (2, 1, 0) 100 (2, 1, 0) (2, 0, 1) 4 6 45 &1_01 0 1 45 (0, 1, 2) (0, 2, 1) 57 (0, 2, 1) (0, 1, 2) 75 (1, 0, 2) (1, 2, 0) 89 (1, 2, 0) (1, 0, 2) 99 (2, 0, 1) (2, 1, 0) 101 (2, 1, 0) (2, 0, 1) 4 6 46 &3_12 1 2 46 (0, 1, 2) (0, 2, 1) 58 (0, 2, 1) (0, 1, 2) 78 (1, 0, 2) (1, 2, 0) 92 (1, 2, 0) (1, 0, 2) 114 (2, 0, 1) (2, 1, 0) 116 (2, 1, 0) (2, 0, 1) 5 6 47 &3_12 1 2 47 (0, 1, 2) (0, 2, 1) 59 (0, 2, 1) (0, 1, 2) 79 (1, 0, 2) (1, 2, 0) 93 (1, 2, 0) (1, 0, 2) 115 (2, 0, 1) (2, 1, 0) 117 (2, 1, 0) (2, 0, 1)

There is only one 3-ary clan with three:

berusa

W C 151 5 1 151 &2_2 2 151 (2, 0, 1) (2, 1, 0) (1, 0, 2) (1, 2, 0) (0, 1, 2) (0, 2, 1) 5 3 158 &4_13 1 3 158 (1, 0, 2) (0, 1, 2) (1, 2, 0) (0, 2, 1) (2, 0, 1) (2, 1, 0) 182 (2, 0, 1) (0, 2, 1) (2, 1, 0) (0, 1, 2) (1, 0, 2) (1, 2, 0) 214 (2, 1, 0) (1, 2, 0) (2, 0, 1) (1, 0, 2) (0, 1, 2) (0, 2, 1) 6 3 189 &2_02 0 2 231 (2, 0, 1) (2, 1, 0) (0, 2, 1) (1, 2, 0) (0, 1, 2) (1, 0, 2) 219 (1, 0, 2) (1, 2, 0) (0, 1, 2) (2, 1, 0) (0, 2, 1) (2, 0, 1) 189 (0, 1, 2) (0, 2, 1) (1, 0, 2) (2, 0, 1) (1, 2, 0) (2, 1, 0) 5 1 233 &1_1 1 233 (0, 1, 2) (0, 2, 1) (1, 0, 2) (1, 2, 0) (2, 0, 1) (2, 1, 0)

Eight are an extreme case. This is very rare:

fidele

W C 1650 4686 5166 6 12 1650 &10_01234 0 1 2 3 4 1650 (3, 1, 0, 2) (1, 3, 2, 0) (1, 2, 3, 0) (3, 0, 1, 2) (2, 1, 0, 3) (0, 3, 2, 1) (2, 0, 1, 3) (0, 2, 3, 1) (0, 2, 3, 1) (2, 0, 1, 3) (0, 3, 2, 1) (2, 1, 0, 3) (3, 0, 1, 2) (1, 2, 3, 0) (3, 1, 0, 2) (1, 3, 2, 0) 1652 (3, 0, 1, 2) (0, 3, 2, 1) (0, 2, 3, 1) (3, 1, 0, 2) (2, 0, 1, 3) (1, 3, 2, 0) (2, 1, 0, 3) (1, 2, 3, 0) (1, 2, 3, 0) (2, 1, 0, 3) (1, 3, 2, 0) (2, 0, 1, 3) (3, 1, 0, 2) (0, 2, 3, 1) (3, 0, 1, 2) (0, 3, 2, 1) 1890 (2, 1, 0, 3) (1, 2, 3, 0) (1, 3, 2, 0) (2, 0, 1, 3) (3, 1, 0, 2) (0, 2, 3, 1) (0, 3, 2, 1) (3, 0, 1, 2) (3, 0, 1, 2) (0, 3, 2, 1) (0, 2, 3, 1) (3, 1, 0, 2) (2, 0, 1, 3) (1, 3, 2, 0) (2, 1, 0, 3) (1, 2, 3, 0) 1892 (2, 0, 1, 3) (0, 2, 3, 1) (0, 3, 2, 1) (2, 1, 0, 3) (3, 0, 1, 2) (1, 2, 3, 0) (1, 3, 2, 0) (3, 1, 0, 2) (3, 1, 0, 2) (1, 3, 2, 0) (1, 2, 3, 0) (3, 0, 1, 2) (2, 1, 0, 3) (0, 3, 2, 1) (2, 0, 1, 3) (0, 2, 3, 1) 4686 (3, 2, 0, 1) (2, 3, 1, 0) (2, 1, 3, 0) (3, 0, 2, 1) (1, 2, 0, 3) (0, 3, 1, 2) (1, 0, 2, 3) (0, 1, 3, 2) (0, 1, 3, 2) (1, 0, 2, 3) (0, 3, 1, 2) (1, 2, 0, 3) (3, 0, 2, 1) (2, 1, 3, 0) (3, 2, 0, 1) (2, 3, 1, 0) 4700 (0, 3, 1, 2) (3, 0, 2, 1) (0, 1, 3, 2) (3, 2, 0, 1) (1, 0, 2, 3) (2, 3, 1, 0) (1, 2, 0, 3) (2, 1, 3, 0) (2, 1, 3, 0) (1, 2, 0, 3) (2, 3, 1, 0) (1, 0, 2, 3) (3, 2, 0, 1) (0, 1, 3, 2) (0, 3, 1, 2) (3, 0, 2, 1) 4938 (1, 2, 0, 3) (2, 1, 3, 0) (1, 0, 2, 3) (2, 3, 1, 0) (0, 1, 3, 2) (3, 2, 0, 1) (0, 3, 1, 2) (3, 0, 2, 1) (3, 0, 2, 1) (0, 3, 1, 2) (3, 2, 0, 1) (0, 1, 3, 2) (2, 3, 1, 0) (1, 0, 2, 3) (1, 2, 0, 3) (2, 1, 3, 0) 4952 (1, 0, 2, 3) (0, 1, 3, 2) (0, 3, 1, 2) (1, 2, 0, 3) (3, 0, 2, 1) (2, 1, 3, 0) (2, 3, 1, 0) (3, 2, 0, 1) (3, 2, 0, 1) (2, 3, 1, 0) (2, 1, 3, 0) (3, 0, 2, 1) (1, 2, 0, 3) (0, 3, 1, 2) (1, 0, 2, 3) (0, 1, 3, 2) 5166 (2, 3, 0, 1) (3, 2, 1, 0) (2, 0, 3, 1) (3, 1, 2, 0) (0, 2, 1, 3) (1, 3, 0, 2) (0, 1, 2, 3) (1, 0, 3, 2) (1, 0, 3, 2) (0, 1, 2, 3) (1, 3, 0, 2) (0, 2, 1, 3) (3, 1, 2, 0) (2, 0, 3, 1) (2, 3, 0, 1) (3, 2, 1, 0) 5178 (1, 3, 0, 2) (3, 1, 2, 0) (1, 0, 3, 2) (3, 2, 1, 0) (0, 1, 2, 3) (2, 3, 0, 1) (0, 2, 1, 3) (2, 0, 3, 1) (2, 0, 3, 1) (0, 2, 1, 3) (2, 3, 0, 1) (0, 1, 2, 3) (3, 2, 1, 0) (1, 0, 3, 2) (1, 3, 0, 2) (3, 1, 2, 0) 5420 (0, 2, 1, 3) (2, 0, 3, 1) (0, 1, 2, 3) (2, 3, 0, 1) (1, 0, 3, 2) (3, 2, 1, 0) (1, 3, 0, 2) (3, 1, 2, 0) (3, 1, 2, 0) (1, 3, 0, 2) (3, 2, 1, 0) (1, 0, 3, 2) (2, 3, 0, 1) (0, 1, 2, 3) (0, 2, 1, 3) (2, 0, 3, 1) 5432 (0, 1, 2, 3) (1, 0, 3, 2) (0, 2, 1, 3) (1, 3, 0, 2) (2, 0, 3, 1) (3, 1, 2, 0) (2, 3, 0, 1) (3, 2, 1, 0) (3, 2, 1, 0) (2, 3, 0, 1) (3, 1, 2, 0) (2, 0, 3, 1) (1, 3, 0, 2) (0, 2, 1, 3) (0, 1, 2, 3) (1, 0, 3, 2) 7 12 1651 &6_123 1 2 3 1651 (2, 1, 0, 3) (0, 3, 2, 1) (1, 2, 3, 0) (3, 0, 1, 2) (2, 0, 1, 3) (0, 2, 3, 1) (3, 1, 0, 2) (1, 3, 2, 0) (1, 3, 2, 0) (3, 1, 0, 2) (2, 0, 1, 3) (0, 2, 3, 1) (3, 0, 1, 2) (1, 2, 3, 0) (0, 3, 2, 1) (2, 1, 0, 3) 1653 (2, 0, 1, 3) (1, 3, 2, 0) (0, 2, 3, 1) (3, 1, 0, 2) (2, 1, 0, 3) (1, 2, 3, 0) (3, 0, 1, 2) (0, 3, 2, 1) (0, 3, 2, 1) (3, 0, 1, 2) (2, 1, 0, 3) (1, 2, 3, 0) (3, 1, 0, 2) (0, 2, 3, 1) (1, 3, 2, 0) (2, 0, 1, 3) 1891 (3, 1, 0, 2) (0, 2, 3, 1) (1, 3, 2, 0) (2, 0, 1, 3) (3, 0, 1, 2) (0, 3, 2, 1) (2, 1, 0, 3) (1, 2, 3, 0) (1, 2, 3, 0) (2, 1, 0, 3) (3, 0, 1, 2) (0, 3, 2, 1) (2, 0, 1, 3) (1, 3, 2, 0) (0, 2, 3, 1) (3, 1, 0, 2) 1893 (3, 0, 1, 2) (1, 2, 3, 0) (0, 3, 2, 1) (2, 1, 0, 3) (3, 1, 0, 2) (1, 3, 2, 0) (2, 0, 1, 3) (0, 2, 3, 1) (0, 2, 3, 1) (2, 0, 1, 3) (3, 1, 0, 2) (1, 3, 2, 0) (2, 1, 0, 3) (0, 3, 2, 1) (1, 2, 3, 0) (3, 0, 1, 2) 4687 (1, 2, 0, 3) (0, 3, 1, 2) (2, 1, 3, 0) (3, 0, 2, 1) (1, 0, 2, 3) (0, 1, 3, 2) (3, 2, 0, 1) (2, 3, 1, 0) (2, 3, 1, 0) (3, 2, 0, 1) (1, 0, 2, 3) (0, 1, 3, 2) (3, 0, 2, 1) (2, 1, 3, 0) (0, 3, 1, 2) (1, 2, 0, 3) 4701 (2, 3, 1, 0) (1, 0, 2, 3) (3, 2, 0, 1) (0, 1, 3, 2) (1, 2, 0, 3) (2, 1, 3, 0) (3, 0, 2, 1) (0, 3, 1, 2) (0, 3, 1, 2) (3, 0, 2, 1) (1, 2, 0, 3) (2, 1, 3, 0) (0, 1, 3, 2) (3, 2, 0, 1) (1, 0, 2, 3) (2, 3, 1, 0) 4939 (3, 2, 0, 1) (0, 1, 3, 2) (2, 3, 1, 0) (1, 0, 2, 3) (0, 3, 1, 2) (3, 0, 2, 1) (1, 2, 0, 3) (2, 1, 3, 0) (2, 1, 3, 0) (1, 2, 0, 3) (0, 3, 1, 2) (3, 0, 2, 1) (1, 0, 2, 3) (2, 3, 1, 0) (0, 1, 3, 2) (3, 2, 0, 1) 4953 (3, 0, 2, 1) (2, 1, 3, 0) (0, 3, 1, 2) (1, 2, 0, 3) (3, 2, 0, 1) (2, 3, 1, 0) (1, 0, 2, 3) (0, 1, 3, 2) (0, 1, 3, 2) (1, 0, 2, 3) (3, 2, 0, 1) (2, 3, 1, 0) (1, 2, 0, 3) (0, 3, 1, 2) (2, 1, 3, 0) (3, 0, 2, 1) 5167 (1, 3, 0, 2) (0, 2, 1, 3) (3, 1, 2, 0) (2, 0, 3, 1) (0, 1, 2, 3) (1, 0, 3, 2) (3, 2, 1, 0) (2, 3, 0, 1) (2, 3, 0, 1) (3, 2, 1, 0) (0, 1, 2, 3) (1, 0, 3, 2) (2, 0, 3, 1) (3, 1, 2, 0) (0, 2, 1, 3) (1, 3, 0, 2) 5179 (2, 3, 0, 1) (0, 1, 2, 3) (3, 2, 1, 0) (1, 0, 3, 2) (0, 2, 1, 3) (2, 0, 3, 1) (3, 1, 2, 0) (1, 3, 0, 2) (1, 3, 0, 2) (3, 1, 2, 0) (0, 2, 1, 3) (2, 0, 3, 1) (1, 0, 3, 2) (3, 2, 1, 0) (0, 1, 2, 3) (2, 3, 0, 1) 5421 (3, 2, 1, 0) (1, 0, 3, 2) (2, 3, 0, 1) (0, 1, 2, 3) (1, 3, 0, 2) (3, 1, 2, 0) (0, 2, 1, 3) (2, 0, 3, 1) (2, 0, 3, 1) (0, 2, 1, 3) (1, 3, 0, 2) (3, 1, 2, 0) (0, 1, 2, 3) (2, 3, 0, 1) (1, 0, 3, 2) (3, 2, 1, 0) 5433 (3, 1, 2, 0) (2, 0, 3, 1) (1, 3, 0, 2) (0, 2, 1, 3) (2, 3, 0, 1) (3, 2, 1, 0) (0, 1, 2, 3) (1, 0, 3, 2) (1, 0, 3, 2) (0, 1, 2, 3) (2, 3, 0, 1) (3, 2, 1, 0) (0, 2, 1, 3) (1, 3, 0, 2) (2, 0, 3, 1) (3, 1, 2, 0)

(This table is wider than most screens. The hover tables on the right can be seen by keeping the mouse on them, and pressing the right arrow key.)