symmetric difference

**1234567** · January 26th 2009, 01:22 PM

can some 1 please help me.

$symmetric difference-untitled.jpg$

**Moo** · January 26th 2009, 11:41 PM

Hello,

Originally Posted by 1234567

can some 1 please help me.

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

It is pretty obvious that it is commutative, by commutativity of $\cap$ and $\cup$ .
As for associativity, you have to prove that $(A \Delta B) \Delta C=A \Delta (B \Delta C)$ . Just use the definition.
What is the identity, that is to say what is E such that $A \Delta E=A$ ? $A \Delta E=(A \cup E) \backslash (A \cap E)$ . If $A \cap E=\emptyset$ and $A \cup E=A$ , you're done. The set E verifying this condition is $E=\emptyset$

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

For the last question, just rewrite the definition :
$(A \cup B) \backslash (A \cap B)=(A \cup B) \cap (A \cap B)^c$ (the complement)

**1234567** · January 27th 2009, 10:15 AM

Originally Posted by Moo

Hello,

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

It is pretty obvious that it is commutative, by commutativity of $\cap$ and $\cup$ .
As for associativity, you have to prove that $(A \Delta B) \Delta C=A \Delta (B \Delta C)$ . Just use the definition.
What is the identity, that is to say what is E such that $A \Delta E=A$ ? $A \Delta E=(A \cup E) \backslash (A \cap E)$ . If $A \cap E=\emptyset$ and $A \cup E=A$ , you're done. The set E verifying this condition is $E=\emptyset$

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

For the last question, just rewrite the definition :
$(A \cup B) \backslash (A \cap B)=(A \cup B) \cap (A \cap B)^c$ (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.

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