# Basic Set Theory

• Aug 24th 2011, 04:22 PM
Aryth
Basic Set Theory
Prove the following for all sets A, B, and C:

$\displaystyle A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$

and

$\displaystyle A\cap B \subset A \subset A \cup C$
• Aug 24th 2011, 04:49 PM
Drexel28
Re: Basic Set Theory
Quote:

Originally Posted by Aryth
Prove the following for all sets A, B, and C:

$\displaystyle A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$

and

$\displaystyle A\cap B \subset A \subset A \cup C$

The first is known as Demorgan's law. Try thinking about it logically, it says that $\displaystyle x\in A\text{ and }(x\notin B\text{ or }x\notin C)\leftrightarrow (x\in A\text{ and }x\notin B)\text{ and }(x\in A\text{ and }x\notin C)$. Can you take it from there?

The second part is fairly standard. Where are you having trouble?
• Aug 24th 2011, 04:56 PM
Aryth
Re: Basic Set Theory
Actually, that's as far as I got on the first one... I did the $\displaystyle \rightarrow$ but I got stumped trying to do the $\displaystyle \leftarrow$ after stating what you did.

On the second one, I guess I just can't see it...

I've never been that good at Set Theory, it slightly confuses me for some reason. I have some time off so I'm trying to study it, and it's not going so well. :/
• Aug 24th 2011, 04:58 PM
Drexel28
Re: Basic Set Theory
Quote:

Originally Posted by Aryth
Actually, that's as far as I got on the first one... I did the $\displaystyle \rightarrow$ but I got stumped trying to do the $\displaystyle \leftarrow$ after stating what you did.

On the second one, I guess I just can't see it...

I've never been that good at Set Theory, it slightly confuses me for some reason. I have some time off so I'm trying to study it, and it's not going so well. :/

Ok. So, let's start with the second one. In sentence form $\displaystyle A\cap B\subseteq A$ says "If $\displaystyle x$ is in $\displaystyle A$ and in $\displaystyle B$ then it is in $\displaystyle A$", obvious now, right? Now try converting the second half.
• Aug 24th 2011, 06:08 PM
Aryth
Re: Basic Set Theory
Quote:

Originally Posted by Drexel28
Ok. So, let's start with the second one. In sentence form $\displaystyle A\cap B\subseteq A$ says "If $\displaystyle x$ is in $\displaystyle A$ and in $\displaystyle B$ then it is in $\displaystyle A$", obvious now, right? Now try converting the second half.

Oh, ok... I see it now. For some reason I wasn't thinking of it like that. Of course if $\displaystyle x$ is in A then it is in A or C.