Studies of Euler diagrams/examples

This is a filtrate of. See here.

A good Euler diagram of this Boolean function needs 3 dimensions.

This representation is inflated. The Boolean function is sufficiently represented by the red-yellow Euler diagram (A, D). The green-blue Venn diagram (B, C) does not add information. But the corresponding circles e.g. in are relevant. So are those in the gap variants Studies of Euler diagrams/vidita and Studies_of_Euler_diagrams/gapspots/hard_and_soft.

This is like shown above, but with the additional information, that B and C are complements:     $$(A \cap D = \varnothing) \and (B^c = C)$$. This is a 4-ary Boolean function, whose bloatless part is the 2-ary. (It is a special case, that the arguments of the bloatless and bloat part are disjoint.)

This representation is also inflated. Only the red-yellow-brown Euler diagram (A, D, E) matters. The green-blue Venn diagram (B, C) does not add information. But the corresponding circles in are relevant.

The green-blue (B, C) and brown-magenta (E, F) Venn diagrams would be separate bundles, if they were not trisected by the red-yellow Euler diagram (A, D). In the graph this is a multiplication, and in the formula it is a conjunction. Compare the filtrates.

Without the small bundle (F, G, H) the surrounding one would be just the red-yellow-brown Euler diagram (A, D, E), as in. But as the small bundle is only in B, and not in C, the green and blue circles are also needed.

If the inner bundle were on its own, it would fall apart into the circles F and H. The circle G would just bisect that Boolean function, adding no information. But in the nested bundle, the circle G is relevant. Removing it would mean, that G implicitly bisects the whole Boolean function (including A...E). But actually it bisects only one cell of the surrounding bundle, namely the one where only B and D intersect.

Compare the gap variants and filtrates.

This is like without spot 9. For the NP equivalence class see NP. For a conversion example see and.