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Math Help - cardinality proof

  1. #1
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    cardinality proof

    How can I prove the following:

    if the cardinality of A1 = the cardinality of A2, the cardinality of B1 = the cardinality of B2, B1 is a subset of A1, and B2 is a subset of A2,
    then the cardinality of (A1 / B1) = the cardinality of (A2 / B2) where / is the set difference operation

    I know there exists a bijective function f: A1 to A2 and a bijective function g: B1 to B2. I need to find a bijective function h: (A1 / B1) to (A2 / B2). Can someone fill in the details?

    Thanks for your help.
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  2. #2
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    Consider this example: B_1  = \left\{ { \cdots ,-6, - 4, - 2} \right\} \subseteq \mathbb{Z}^ -  \;\& \;A_1  = \left\{ {5,6, \cdots } \right\} \subseteq \mathbb{Z}^ +  .
    It is clear that \mathbb{Z}^ - \leftrightarrow \mathbb{Z}^ + .
    Define \Phi :A_1  \leftrightarrow B_1 ,\;\;\Phi (a) =  - 2\left( {a - 4} \right). Prove that \Phi is a bijection.
    Now consider \left( {\mathbb{Z}^ -  \backslash B_1 } \right)\;\& \;\left( {\mathbb{Z}^ +  \backslash A_1 } \right).
    Is your statement true?
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  3. #3
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    Quote Originally Posted by PvtBillPilgrim View Post
    How can I prove the following:

    if the cardinality of A1 = the cardinality of A2, the cardinality of B1 = the cardinality of B2, B1 is a subset of A1, and B2 is a subset of A2,
    then the cardinality of (A1 / B1) = the cardinality of (A2 / B2) where / is the set difference operation

    I know there exists a bijective function f: A1 to A2 and a bijective function g: B1 to B2. I need to find a bijective function h: (A1 / B1) to (A2 / B2). Can someone fill in the details?

    Thanks for your help.
    Have the courtesy to edit your posts accordingly if you post the exact same questions at other websites. Especially if you get replies.

    Or don't you care that what you're doing has the potential to waste the time of people who are giving you free help in good faith.
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