1
$\begingroup$

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:

enter image description here

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.

$\endgroup$

2 Answers 2

1
$\begingroup$

For each set (circle) the region you care about is either inside it or outside it.

Therefore the region shown in your diagram for sets ABCDE corresponds to binary word 01101.

A simple program that generates all such regions is just counting in binary from $0$ to $2^n$ where $n$ the number of sets.

enter image description here

You can perform operations with binary numbers

enter image description here

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .