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Math Help - Inverses With Respect To Union and Intersection

  1. #1
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    Inverses With Respect To Union and Intersection

    I'm not sure about my solution to the following problem and I'd appreciate it if someone could check it.

    ---

    Let S(\mathbb{Z}) be the set of all subsets of \mathbb{Z}.
    a. Which subsets A of \mathbb{Z} have inverses for \cup? What are they?

    b. Which subsets A of \mathbb{Z} have inverses for \cap? What are they?
    ---

    Here is my solution:

    a. \{\} is the identity of S(\mathbb{Z}) with respect to \cup so the only subset A of \mathbb{Z} with an inverse with respect to \cup is \{\}, namely itself.

    b. \mathbb{Z} is the identity of S(\mathbb{Z}) with respect to \cap so the only subset A of \mathbb{Z} with an inverse with respect to \cap is \mathbb{Z}, namely itself.
    Last edited by rualin; July 6th 2007 at 03:14 PM.
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  2. #2
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    Quote Originally Posted by rualin View Post
    I'm not sure about my solution to the following problem and I'd appreciate it if someone could check it.

    ---

    Let S(\mathbb{Z}) be the set of all subsets of \mathbb{Z}.
    a. Which subsets A of \mathbb{Z} have inverses for \cup? What are they?

    b. Which subsets A of \mathbb{Z} have inverses for \cap? What are they?
    ---

    Here is my solution:

    a. \{\} is the identity of S(\mathbb{Z}) with respect to \cup so the only subset A of \mathbb{Z} with an inverse with respect to \cup is \{\}, namely itself.

    b. \mathbb{Z} is the identity of S(\mathbb{Z}) with respect to \cap so the only subset A of \mathbb{Z} with an inverse with respect to \cap is \mathbb{Z}, namely itself.
    Good Job.
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