Prove the following for all sets A, B, and C:

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

and

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

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- Aug 24th 2011, 04:22 PMArythBasic 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 PMDrexel28Re: Basic Set Theory
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 PMArythRe: 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 PMDrexel28Re: Basic Set Theory
- Aug 24th 2011, 06:08 PMArythRe: Basic Set Theory