This Demonstration shows all 127 nonempty unions and intersections of three sets, , , . You can show or hide the corresponding set notation, given in terms of union, intersection, and complement.

Define a Venn diagram in the plane as a family of nonempty simply connected sets all of whose iintersections are also nonempty simply connected sets. Here is a construction of a Venn diagram of size by induction.

Suppose that the family of sets , , is a Venn diagram and define the boundary of to be . Starting from outside of , the curve intersects the boundaries of the other sets in cyclic order at points , . Thicken the path along from to and slide it back along the path a small distance to form the set . (In the diagram, , , .)