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  (: H and F in G)
 * one set in an otherwise empty intersection  (: C in A∩B)

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

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

(7 of 8 cells)
Below are different Euler diagrams of the 3-ary Boolean function $$A \cap B \cap C = \varnothing$$.

⚒
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.

(2×4) ⚒
If the gapspots were true, the red circle would vanish.

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 ) 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.)

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