# Studies of Euler diagrams/decompose

## Shape of the Euler diagram

 the four cells of bundle AB   (outside of sets C and D) the two cells of bundle C   (inside of set B, outside of A and D) the two cells of bundle D   (outside of sets A, B and C) disjunction of the three conjunctions above

These images represent the 4-ary Boolean function ${\displaystyle (\neg C\land \neg D)\lor (\neg A\land B\land \neg D)\lor (\neg A\land \neg B\land \neg C)}$.
It is true in 6 of 16 cases, and is part of BEC 127 (rows (22,2) and (23,0)).

The images below show the truth table with false places grayed out, the Euler diagram, and the corresponding graph.

The images on the right show the decomposition into three bundles, i.e. parts of the Euler diagram that are connected by crossing circles.

For this particular Boolean function, the colored squares can be seen as true.
The colored truth tables on the left represent the cells of the bundle, i.e. all cells that touch one of its circles.
The gray/white truth tables on the right are always minterms, and represent the position of the bundle in relation to every other set, i.e. inside or outside of it.

## Functions with this Euler diagram shape

Light gray areas of the Euler diagram are empty, i.e. the corresponding fields in the truth table are false. The Euler diagram could not be drawn without these gap cells, because each border can belong to only one set. A border between always corresponds to a change of one bit.

Truth table fields in dark gray do not have corresponding fields in the Euler diagram. (They are the same in each file.)

There seem to be only four Boolean functions with this Euler diagram shape. Many cells could not be empty, without changing the shape of the Euler diagram. E.g. the intersection of A and B can not be empty, because then the two circles would be separate. And the left crescent can not be empty, because then A would be within B. And no set can be empty, because that would make it's complement the universe, which should be displayed as the border of the Euler diagram.

murife
The intersection ¬A ∧ B ∧ ¬C is marked as empty.
Thus the truth table of bundle AB is TTFT and that of the bundle C is FT.     BEC 41, row (5,0)
tefabi
Both cells mentioned above are empty.     BEC 205, rows (16,3) and (17,1)