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My Venn Diagram Mistake

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Background

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

A 4-set 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.

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