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 ⚒

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   (karafa: C in A∩B)

XOR ⚒

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

3-ary
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

vanatu ⚒

This is a XOR including an OR:    $A~\oplus ~(B\lor C)~\oplus ~D$           It is a gap variant of vidita and a filtrate of tomute.           The gapspots 2, 4, 6 are hard; 1 and 8 are optional.

with optional gapspots

only hard gapspots
2D Euler diagram with separated gap cell 6	3D Euler diagram of vidita
	gap cells of vanatu

bunese (7 of 8 cells)

Below are different Euler diagrams of the 3-ary Boolean function $A\cap B\cap C=\varnothing$.

variants
The gap cell is degenerated to a single point, where three borders cross. Within this project, this approach is not used. This is the most obvious representation: The sets are represented by the inside of circles, and their intersection is the gap cell. Here the sets are represented by the outside of the circles, so that the gap cell is on the outside. With three straight lines the plane can be partitioned in seven areas. This is essentially the representation on the left with the border moved inside.

dagoro ⚒

The gaps 2 and 6 are hard. Gap 0 is optional. Without C this bundle falls apart in two bundles with a gap cell between them.

variants

foravo (hexagon)

variants
This function can reasonably be shown with two or no gapspots. The latter means, that the three borders intersect in one point. This point could be enlarged into gapspot 0 or 7. But that would be a random choice, which should be avoided.
both gapspots	gapspot 7	no gapspots	gapspot 0

2×3

grid
potula	kinide ⚒	gilipi ⚒	gelade

redrawn

ternary labels

kagusi (2×4) ⚒

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

graph, Euler diagram

rather bad 1D Euler diagram
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

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.)

multi-bundle 3-2-2-1   todeda	5×4   gufaro
Distance between cells 37 and 32:   2 above, 4 below	Distance between cells 10 and 24:   2 above, 6 below

piferi

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

matrix and circular graph
3×3 graph with gapspot	1-dimensional graph and Euler diagram closed to a circle

Euler diagram (matrix and cylinder)

5×5 circles

kabine

kasete