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Math Help - Cardinality properties

  1. #1
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    Cardinality properties

    Hello,

     A_{1} and  B_{1} have the same cardinality, so do  A_{2} and  B_{2}

    I need to check if the following sets also have the same cardinality:

    1)  A_{1} \times A_{2} and  B_{1} \times B_{2} .

    Those two cartesian products have the same cardinality, as it's easy to construct a bijection between those two sets using bijections  f:A_{1} \rightarrow B_{1} and  f:A_{2} \rightarrow B_{2}.

    2)  A_{1} \cap A_{2} and  B_{1} \cap B_{2} .

    Counterexample:

    A_{1}={1,2,3}
    A_{2}={1,2}
    B_{1}={4,5,6}
    B_{2}={4,10}

    Then  |A_{1} \cap A_{2}|=2 and  |B_{1} \cap B_{2} |=1.
    2)  A_{1} \cap A_{2} and  B_{1} \cap B_{2} .

    Counterexample:

    A_{1}={1,2,3}
    A_{2}={1,2}
    B_{1}={4,5,6}
    B_{2}={4,10}

    Then  |A_{1} \cup A_{2}|=3 and  |B_{1} \cup B_{2} |=4.
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  2. #2
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    Re: Cardinality properties

    Quote Originally Posted by MachinePL1993 View Post
     A_{1} and  B_{1} have the same cardinality, so do  A_{2} and  B_{2}
    I need to check if the following sets also have the same cardinality:
    1)  A_{1} \times A_{2} and  B_{1} \times B_{2} .
    Those two cartesian products have the same cardinality, as it's easy to construct a bijection between those two sets using bijections  f:A_{1} \rightarrow B_{1} and  f:A_{2} \rightarrow B_{2}.

    2)  A_{1} \cap A_{2} and  B_{1} \cap B_{2} .

    Counterexample:

    A_{1}={1,2,3}
    A_{2}={1,2}
    B_{1}={4,5,6}
    B_{2}={4,10}

    Then  |A_{1} \cap A_{2}|=2 and  |B_{1} \cap B_{2} |=1.
    2)  A_{1} \cap A_{2} and  B_{1} \cap B_{2} .

    Counterexample:

    A_{1}={1,2,3}
    A_{2}={1,2}
    B_{1}={4,5,6}
    B_{2}={4,10}

    Then  |A_{1} \cup A_{2}|=3 and  |B_{1} \cup B_{2} |=4.

    You have several notation mistakes. It should be
    construct a bijection between those two sets using bijections  f:A_{1} \rightarrow B_{1} and  g:A_{2} \rightarrow B_{2}.

    And
    A_{1}={1,2,3}
    A_{2}={1,2}
    B_{1}={4,5,6}
    B_{2}={4,10}

    Then  |A_{1} \cap A_{2}|=2 and  |B_{1} \cap B_{2} |=1.
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