Set Theory Help.

**Kostapoulous** · April 7th 2011, 11:19 AM

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.

**Plato** · April 7th 2011, 11:28 AM

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 $A=B~?$ .

**emakarov** · April 7th 2011, 11:35 AM

$A\vartriangle B=\emptyset$ iff $A = B$ .

$A\vartriangle B=U$ iff $A\cup B = U$ and $A\cap B=\emptyset$ .

**Kostapoulous** · April 7th 2011, 11:35 AM

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.

**Plato** · April 7th 2011, 11:40 AM

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 $A\Delta B=A\cup B$

**Kostapoulous** · April 7th 2011, 12:38 PM

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?

**Plato** · April 7th 2011, 12:48 PM

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?

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

$A\Delta B\ne\emptyset{\text{ if and only if }A<br /> \ne B.$

**Kostapoulous** · April 7th 2011, 12:59 PM

Originally Posted by Plato

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

$A\Delta B\ne\emptyset{\text{ if and only if }A<br /> \ne B.$

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

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