# Annoying set theory question

• Apr 18th 2013, 12:43 PM
cac2008
Annoying set theory question
**apologies if this is the wrong part of the forum... it was the closest I could find :) **

Whilst doing questions in preparation for uni this year, I've been working on some abstract algebra. I was doing pretty well until I came across this:

$P=(A \cap B) \cup (C \cap D)$
$Q=(A \cup B) \cap (C \cup D)$
Prove that $P \subseteq Q$

It looks like it should be easy but it's been resisting all attempts that I've tried. I've tried a direct proof that $x \in P \Rightarrow x \in Q$ but that didn't go anywhere, nor did a contradiction.

Can anyone please make me look a fool and pull out a solution that is from THE BOOK? (Nerd)
• Apr 18th 2013, 01:02 PM
Plato
Re: Annoying set theory question
Quote:

Originally Posted by cac2008
$P=(A \cap B) \cup (C \cap D)$
$Q=(A \cup B) \cap (C \cup D)$
Prove that $P \subseteq Q$

Let $A=\{1,2\},~B=\{2,3\},~\&~C=\{3,4\},~D=\{4,5,\},~$

Is it true?
• Apr 18th 2013, 01:16 PM
cac2008
Re: Annoying set theory question
Quote:

Originally Posted by Plato
Let $A=\{1,2\},~B=\{2,3\},~\&~C=\{3,4\},~D=\{4,5,\},~$

Is it true?

With those, P={2,4} and Q={3} so no, it isn't.

Just done a Venn diagram (why did I not do that earlier!?) and it looks like the question is written wrong as:
Attachment 28009
Attachment 28010

Please tell me that this is not a common thing with Uni textbooks!
• Apr 19th 2013, 03:54 AM
Draco93
Re: Annoying set theory question
Plato has given a counter example to the theorem, which evidently disproves the theorem. If the question doesn't place constraints stating that all the sets have to be different,
we could try making simpler counter examples revolving around the empty set.

A = B = {1}
C = D = ∅

Then, {1} is a subset of ∅ will obviously be false.