A Venn diagram shows all possible logical relations between a finite collection of different sets. Below is a likely familiar example of a Venn diagram with 2 and 3 sets, respectively:

How might you extend this to larger and larger numbers of sets?

Here’s my solution:

Let’s start with a Venn diagram for 0 sets. We can represent our single possibility (no sets) with a single (empty) region - this square:

Next, we add one set. Let’s call this set Purple. We now have two regions, one for no sets, and one for just Purple:

Naturally, we add another set next (Green). We now have 4 regions, corresponding to no sets, just Purple, just Green, and Purple & Green:

Next, we add Blue. We now have 8 regions, corresponding to no sets, just Purple, just Green, and Purple & Green, just Blue, Purple & Blue, Green & Blue, and Purple & Green & Blue.

Notice that every time we add a new set, our number of subsets (regions) doubles. We had 4 subsets before adding Blue. When we add Blue, for each of the 4 previous subsets, we can choose to either add Blue to that subset or not.

Next, we add Pink, and we have 16 regions: