# Eulero-Venn diagrams creator/calculator

The other day I was having some relax and I found a challenge online about set theory. The question was: "how to descrive the black-shaded area $$H$$ in terms of the other sets, using the elementary set operations $$\cup, \cap, \backslash, \triangle$$ (where $$\triangle$$ is the symmetric difference)? Call $$\Omega$$ the Universe if necessary.

This was the photo:

Questions:

• Is there a program, online, in which one can put those kind of draws and determine all the possible sub-regions (not only the black shaded one)?

• Minding about the question, I have come up with two different solutions, but I don't know if they are correct for I'm exhausted.

First one:

$$H = (D \backslash E) \cap (B\backslash A)$$

Second one:

$$((B \cap C \cap D) \backslash A) \backslash E$$

Thank you for any help an answer.

A simple program that generates all such regions is just counting in binary from $$0$$ to $$2^n$$ where $$n$$ the number of sets.
I think, $$((D \cap C)\cap(E))\cap(A')$$ also works, where $$A'$$ is the relative complement of $$A$$ in $$B$$ Or more formally, $$B/A$$ Also, I think your answers are correct.