Lex and Gehlenborg reply:

We thank Lentini and Habtemariam1 for their thoughtful comments on our article regarding the formula for the number of possible intersections of n sets.

The formula used in our original article2 implicitly includes intersections of each of the n sets with the universe (set of all objects one wants to consider and that may or may not be included in the n sets) as well as the set of elements in the universe that are not included in any of the n sets. The latter set indeed does not conform to the usual definition of an intersection, but for the sake of brevity we chose not to introduce an additional term to refer to both this set and the intersections. Further confusion may have arisen from the fact that the shape representing the universe is often not explicitly drawn in Venn and Euler diagrams, as in our article2 or in the comment by Lentini and Habtemariam. Figure 1 shows an example of a Venn diagram with two sets and an explicit representation of the universe.

Figure 1
figure 1

A Venn diagram showing the intersections of sets A and B with each other and the implicit universe U, which is shown explicitly in this illustration.

Full size image

Finally, we want to emphasize that the exponential growth of the number of possible intersections of n sets is the primary visualization challenge in this context and a key motivation for the development of interactive techniques such as UpSet3.