# Set Theory Help.

• Apr 7th 2011, 12:19 PM
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
$Z = (A \cap ¬B) \cup (¬A \cap B)$

Therefore, $Z = A \Delta B$

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

Thanks for any help.
• Apr 7th 2011, 12:28 PM
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
$Z = (A \cap ¬B) \cup (¬A \cap B)$
Therefore, $Z = A \Delta B$
I need to prove Z could equal Ø.
Also, Z can equal U.

What if $A=B~?$.
• Apr 7th 2011, 12:35 PM
emakarov
$A\vartriangle B=\emptyset$ iff $A = B$.

$A\vartriangle B=U$ iff $A\cup B = U$ and $A\cap B=\emptyset$.
• Apr 7th 2011, 12:35 PM
Kostapoulous
Quote:

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

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, 12:40 PM
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 $A\ne B$ then $A\Delta B\ne\emptyset.$

If $A\cap B=\emptyset$ then $A\Delta B=A\cup B$
• Apr 7th 2011, 01:38 PM
Kostapoulous
Quote:

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

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

But if Z = $A\Delta B$ , 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, 01:48 PM
Plato
Quote:

Originally Posted by Kostapoulous
But if Z = $A\Delta B$ , 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?

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

$A\Delta B\ne\emptyset{\text{ if and only if }A
\ne B.$
• Apr 7th 2011, 01:59 PM
Kostapoulous
Quote:

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

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

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