A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. (https://en.m.wikipedia.org/wiki/Venn_diagram)
It is easy to make a Venn diagram for three sets. Just draw three circles as shown in the figure. For four sets, it is getting more difficult. There are lot of possible ways making a Venn diagram for 4 sets (just check the wikipedia or imagine with four ellipses), but it is impossible to make it with four circles. But why?
I only know that it is impossible. I can prove it but my solution is really complicated (maybe wrong as well), but I think there might be an easy, beautiful solution.
Please help! Thanks in advance!
Best Answer
Two circles can intersect in at most two points. If you have $n-1$ circles, then adding an $n$-th will introduce at most $2n-2$ new intersections, and so the number of regions can increase by at most $2n-2$.
With three circles, one has at most eight regions (why?). Adding a fourth will increase the number of regions by at most six, so we get at most fourteen regions. Not enough.