# Gentle sets of Boolean functions

Ж :
* ANF   matrix   twins
* noble   gentle
* smallest Zhegalkin index
It is possible two extend the twin relationship to sets of Boolean functions. Two sets $A,B$ are twins, iff $a\in A\implies \operatorname {twin} (a)\in B$. Sets that are their own twins shall be called gentle. (A set of nobles is gentle, but a gentle set does not need to contain nobles.)
Transposing the matrix positions of Zhegalkin indices often leads to gentle sets. The upper row in the following table shows some sets of Zhegalkin indices. (The tt links lead to the corresponding truth tables.) The lower row shows the transposed positions. Fields marked with the same color form gentle sets.
| super-clan (tt) | tribes (tt) | quaestor 10 (tt) |
| transposed super-clan | transposed tribes | patron 72 |
The following table shows the eight Boolean functions marked pink in the transposed super-clan. Twins are shown below each other, and the corresponding patrons are shown in the third row.
| 9 | 7 | 11 | 13 |
| 119 | 153 | 221 | 187 |
| 126 | 158 | 214 | 182 |

## gentle factions

The table below shows representative Zhegalkin indices of gentle factions in their respective quadrants. Representatives of noble factions are bold. Odd gentle factions (blue and green) seem to exist only for even arities.
| arity | total size | size | | evil | odious |
| 1 | 2 | 1 | even | 0 | 2 |
| 2 | 6 | 2 | even | 0, 6 | 8, 14 |
| 2 | 6 | 1 | odd | 3 | 11 |
| 3 | 12 | 6 | even | 0, 10, 30, 40, 46, 126 | 128, 138, 158, 168, 174, 254 |
| 4 | 224 | 76 | even | 0, 60, 136, 238, 510, 680, 718, 854, 874, 918, 938, 940, 990, 1632, 1678, 1690, 1696, 1768, 1950, 1954, 2176, 2216, 2278, 2454, 2474, 2752, 2770, 2812, 2992, 3030, 3070, 3808, 3826, 4086, 5760, 6014, 6018, 6078, 6280, 6298, 6304, 6324, 6558, 6648, 6830, 6856, 6876, 7134, 7138, 7158, 7912, 8190, 10432, 10492, 10654, 10732, 10750, 10920, 10926, 10952, 10958, 11464, 11470, 11482, 11508, 11766, 12286, 15572, 15592, 15830, 26752, 26812, 27606, 27626, 28384, 32766 | 32768, 32828, 32904, 33006, 33278, 33448, 33486, 33622, 33642, 33686, 33706, 33708, 33758, 34400, 34446, 34458, 34464, 34536, 34718, 34722, 34944, 34984, 35046, 35222, 35242, 35520, 35538, 35580, 35760, 35798, 35838, 36576, 36594, 36854, 38528, 38782, 38786, 38846, 39048, 39066, 39072, 39092, 39326, 39416, 39598, 39624, 39644, 39902, 39906, 39926, 40680, 40958, 43200, 43260, 43422, 43500, 43518, 43688, 43694, 43720, 43726, 44232, 44238, 44250, 44276, 44534, 45054, 48340, 48360, 48598, 59520, 59580, 60374, 60394, 61152, 65534 |
| 4 | 224 | 36 | odd | 15, 175, 857, 907, 921, 1005, 1647, 1683, 1711, 1725, 2227, 2457, 2469, 2767, 2803, 3045, 4089, 6027, 6279, 6331, 6573, 6907, 7121, 7149, 7911, 10447, 10669, 10699, 10991, 11477, 11517, 11769, 15579, 26767, 27609, 28399 | 32783, 32943, 33625, 33675, 33689, 33773, 34415, 34451, 34479, 34493, 34995, 35225, 35237, 35535, 35571, 35813, 36857, 38795, 39047, 39099, 39341, 39675, 39889, 39917, 40679, 43215, 43437, 43467, 43759, 44245, 44285, 44537, 48347, 59535, 60377, 61167 |

### 3-ary gentle great factions

There are 20 great factions with arity 3. The 6 gentle ones among them are shown below. The even factions in them (with red and yellow borders) are gentle, while the odd ones (with blue and green borders) are twins. 4 of the 12 gentle factions are not noble.