# Set Theory Help.

• Apr 7th 2011, 11:19 AM
Kostapoulous
Set Theory Help.
Ok guys, for any 2 non-empty sets X and Y drawn from the same univserse, U, let Z be defined
$\displaystyle Z = (A \cap ¬B) \cup (¬A \cap B)$

Therefore, $\displaystyle Z = A \Delta B$

I need to prove Z could equal Ø.
Also, Z can equal U.

Thanks for any help.
• Apr 7th 2011, 11:28 AM
Plato
Quote:

Originally Posted by Kostapoulous
Ok guys, for any 2 non-empty sets X and Y drawn from the same univserse, U, let Z be defined
$\displaystyle Z = (A \cap ¬B) \cup (¬A \cap B)$
Therefore, $\displaystyle Z = A \Delta B$
I need to prove Z could equal Ø.
Also, Z can equal U.

What if $\displaystyle A=B~?$.
• Apr 7th 2011, 11:35 AM
emakarov
$\displaystyle A\vartriangle B=\emptyset$ iff $\displaystyle A = B$.

$\displaystyle A\vartriangle B=U$ iff $\displaystyle A\cup B = U$ and $\displaystyle A\cap B=\emptyset$.
• Apr 7th 2011, 11:35 AM
Kostapoulous
Quote:

Originally Posted by Plato
What if $\displaystyle A=B~?$.

I don't quite follow what you mean, I was under the assumption that A could not equal B. I am quite confused.
• Apr 7th 2011, 11:40 AM
Plato
Quote:

Originally Posted by Kostapoulous
I don't quite follow what you mean, I was under the assumption that A could not equal B. I am quite confused.

If both are non-empty and $\displaystyle A\ne B$ then $\displaystyle A\Delta B\ne\emptyset.$

If $\displaystyle A\cap B=\emptyset$ then $\displaystyle A\Delta B=A\cup B$
• Apr 7th 2011, 12:38 PM
Kostapoulous
Quote:

Originally Posted by Plato
If both are non-empty and $\displaystyle A\ne B$ then $\displaystyle A\Delta B\ne\emptyset.$

If $\displaystyle A\cap B=\emptyset$ then $\displaystyle A\Delta B=A\cup B$

But if Z = $\displaystyle A\Delta B$ , i.e. the symmetrical difference, which is what is in set A but not B, and what is in set B but not A, how can it ever equal Ø if they are non-empty? Would they not always have some elements in the sets?
• Apr 7th 2011, 12:48 PM
Plato
Quote:

Originally Posted by Kostapoulous
But if Z = $\displaystyle A\Delta B$ , i.e. the symmetrical difference, which is what is in set A but not B, and what is in set B but not A, how can it ever equal Ø if they are non-empty? Would they not always have some elements in the sets?

$\displaystyle A\Delta B=\emptyset{\text{ if and only if }A=B.$

$\displaystyle A\Delta B\ne\emptyset{\text{ if and only if }A \ne B.$
• Apr 7th 2011, 12:59 PM
Kostapoulous
Quote:

Originally Posted by Plato
$\displaystyle A\Delta B=\emptyset{\text{ if and only if }A=B.$

$\displaystyle A\Delta B\ne\emptyset{\text{ if and only if }A \ne B.$

Yes! It's finally clicked in my head! I feel like such an idiot now for not picking this up earlier!