can some 1 please help me.

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- Jan 26th 2009, 01:22 PM1234567symmetric difference
can some 1 please help me.

Attachment 9838 - Jan 26th 2009, 11:41 PMMoo
Hello,

$\displaystyle A \Delta B=(A \cup B) \backslash (A \cap B)$, this is according to the definition you're given.

It is pretty obvious that it is commutative, by commutativity of $\displaystyle \cap$ and $\displaystyle \cup$.

As for associativity, you have to prove that $\displaystyle (A \Delta B) \Delta C=A \Delta (B \Delta C)$. Just use the definition.

What is the identity, that is to say what is E such that $\displaystyle A \Delta E=A$ ? $\displaystyle A \Delta E=(A \cup E) \backslash (A \cap E)$. If $\displaystyle A \cap E=\emptyset$ and $\displaystyle A \cup E=A$, you're done. The set E verifying this condition is $\displaystyle E=\emptyset$

As for the inverse, you're looking for F such that $\displaystyle A \Delta F=\emptyset$. $\displaystyle (A \cup F) \backslash (A \cap F)=\emptyset$

If $\displaystyle A \cup F=A \cap F$, then you're done.

So $\displaystyle F=A$

For the last question, just rewrite the definition :

$\displaystyle (A \cup B) \backslash (A \cap B)=(A \cup B) \cap (A \cap B)^c$ (the complement) - Jan 27th 2009, 10:15 AM1234567