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Math Help - symmetric difference

  1. #1
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    symmetric difference

    can some 1 please help me.

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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by 1234567 View Post
    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)
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  3. #3
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    Quote Originally Posted by Moo View Post
    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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