## Background

I tried this Mathematics Stack Exchange question by drawing a Venn Diagram.

## Problem

A downvote was quickly casted on my answer below which a comment about
the non-existence of a Venn diagram with 4 sets was left. I clicked
on the link to another question in that comment to
understand why I got this downvote: **Why a 4-set Venn diagram
doesn’t exist?**

## Explanation

I understood the arguments of Joebot’s answer up to the “application
of binomial theorem”, but I *didn’t* understand the Euler’s formular.
In fact, for any 3-D object with edges, vertices and faces, one can
represent it by a 2-D planar graph, on which the outermost region also
represents one of the faces of the 3-D object. Thus, the Euler’s
formula for any 3-D object can be seen on a 2-D planar graph. I’m
amazed that this can be proved by mathematical induction on the number
of regions on the 2-D graph (which corresponds to that of faces of the
3-D object). IMHO, this should be taught in high schools as examples
of mathematical induction, instead of divisibility of integers, which
can be done better using modular arithmetic.