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Math Help - Sigma Algebras

  1. #1
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    Sigma Algebras

    The problem is:

    Let U={a,b,c}.

    Write the power set and an explicitly list of all sigma algebras of subsets of U.

    I got the power set to be {{empty set},{a},{b}{c},{a,b},{ac},{abc}}

    I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

    Could another one be {{a},{b},{c}}?

    Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?

    Any help on this would be appreciated. Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by taypez View Post
    The problem is:

    Let U={a,b,c}.

    Write the power set and an explicitly list of all sigma algebras of subsets of U.

    I got the power set to be {{empty set},{a},{b},{c},{a,b},{ac},{abc}}

    I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

    Could another one be {{a},{b},{c}}?

    Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?

    Any help on this would be appreciated. Thanks
    i dunno what sigma algebras are, just wanted to point out that your power set is incomplete. you are missing the subset {b,c}
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by taypez View Post
    The problem is:

    Let U={a,b,c}.

    Write the power set and an explicitly list of all sigma algebras of subsets of U.

    I got the power set to be {{empty set},{a},{b}{c},{a,b},{ac},{abc}}

    I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

    Could another one be {{a},{b},{c}}?
    No because the complement of one of your elements \{ a \} ^C = \{ b, c \} is not in your set. Also the countable union \{ a \} \cup \{ b \} = \{ a, b \} is not in your set.

    Quote Originally Posted by taypez View Post
    Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?
    Should be. All the complements and finite unions of each element are in the set.

    -Dan
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