# Studies of Euler diagrams/gapspots/hard and soft

Some gapspots can not be avoided, while others are a design choice. They could be called *hard* and *soft*. On this page functions with hard gapspots are marked with ⚒.

## between different bundles ⚒[edit | edit source]

There is no way to avoid gapspots between different bundles. The simplest examples are:

- two sets in an otherwise empty set (sarina:
*H*and*F*in*G*) - one set in an otherwise empty intersection (cm1:
*C*in*A*and*B*)

bazinga gapspots | ||
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sarina |
futare |
geteso |

bundles 3-2-2-1 | ||
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a1 |
b2 |
cm1 |

## XOR ⚒[edit | edit source]

Functions in the same box are complements. Those on the left (right) are true in cells with even (odd) digit sum.

3-ary | |
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One could choose an Euler diagram with 7 cells (compare bunese), which would remove one of the gapspots. But that would be a random choice, which should be avoided. | |

selera |
pelele |

5x4 | ||
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linaki |
karifu |

bazinga | ||
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kulika |
torova |

## vanatu ⚒[edit | edit source]

This is a XOR including an OR: It is a gap variant of bar and a filtrate of tomute.

3-ary | |
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## bunese (7 of 8 cells)[edit | edit source]

Below are three possible Euler diagrams of the same Boolean function. It is in logic, and in set theory.

variants | ||
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- left: The false cell is shown as a gapspot.
- middle: An intersection of three borders is shown instead.
- right: With three straight lines the plane can be partitioned in seven areas.
| ||

similar example darimi ⚒ | |
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Here the spot 4 is an optional gap, while 6 and 7 are hard. (This is a filtrate of tomute.) | |

## foravo (hexagon)[edit | edit source]

## 2×3[edit | edit source]

grid | |||
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potula |
kinide ⚒ |
gilipi ⚒ |
gelade |

redrawn | |||
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ternary labels |
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## kagusi (2×4) ⚒[edit | edit source]

If the gapspots were true, the red circle would vanish.

graph, Euler diagram | |
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rather bad 1D Euler diagram |
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The problem with this representation is, that it was chosen to remove gapspot 0 rather than 9. But random choices should be avoided. |

## 1- or 2-dimensional[edit | edit source]

Euler diagrams can be drawn with gapspots in a higher dimension, or without in a lower. Generally one should prefer the lowest possible dimension. But it is reasonable to demand from an Euler diagram, that the set borders be contiguous - which a circle in one dimension (a 0-sphere) is not. So in this case, one might prefer the 2D diagrams, and consider the gapspots necessary. (The disconnected left and right parts are easier seen in this version of the 1D diagram on the right.)

gapspots bundles 3-2-2-1 cm2 | gapspots 5×4 gufaro | |

Distance between cells 37 and 32: 2 above, 4 below | Distance between cells 10 and 24: 2 above, 6 below | |

## piferi[edit | edit source]

Like example putuki, but with spot 0 as gapspot. (Compare logota, another octagon.)

matrix and circular graph | |
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Euler diagram (matrix and cylinder) | |
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