Venn diagrams (top), Euler diagrams (middle) and dual graphs of the Euler diagrams (bottom) of relations corresponding to 3-ary Boolean functions
Areas marked black contain no elements. Euler diagrams avoid showing empty areas, but this is not always possible.

See also: v:User:Watchduck/Logic#3-ary relations

In the dual graphs all the vertices above the vertex marked with a black A, B or C belong to that set.
(These naming vertices are marked by a thicker edge.)
Four graphs contain empty naming vertices, shown as small black dots.
(An empty area of an Euler diagram only has a corresponding vertex in the dual graph when it is needed as a naming vertex.)
To make clear which areas and vertices belong together each of them contains small red letters showing to which of the three sets it belongs.

Each function belongs to one of  A000616(3) = 22 big equivalence classes (becs).
Relations from complementary becs are ordered symmetrically to the vertical middle axis of the image - except in rectangle 4: All relations with 4 white areas are from self complementary becs.

Each one of the 256 possible relations can be expressed by one of these Euler diagrams by permuting and negating the arguments A, B, C.
E.g. can be expressed by .

The beige numbers denote the number of white areas in the diagrams in the corresponding beige rectangle.
(That is the number of ones in the corresponding Boolean functions.)

 A039754