1. ## symmetric difference

2. Hello,
Originally Posted by 1234567

$\displaystyle A \Delta B=(A \cup B) \backslash (A \cap B)$, 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)

3. Originally Posted by Moo
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)
THANKS FOR THE HELP, I NOW UNDERSTAND HOW TO DO THE
identity, and inverse laws BUT CAN NOT STILL DO THE