# symmetric difference

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• Jan 26th 2009, 01:22 PM
1234567
symmetric difference
can some 1 please help me.

Attachment 9838
• Jan 26th 2009, 11:41 PM
Moo
Hello,
Quote:

Originally Posted by 1234567
can some 1 please help me.

Attachment 9838

\$\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 AM
1234567
Quote:

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
commutative, associative AND distributive law, CAN YOU PLEASE HELP ME.