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Thread: Cardinality properties

  1. #1
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    Cardinality properties

    Hello,

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

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

    1)$\displaystyle A_{1} \times A_{2} $ and $\displaystyle 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 $\displaystyle f:A_{1} \rightarrow B_{1}$ and $\displaystyle f:A_{2} \rightarrow B_{2}$.

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

    Counterexample:

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

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

    Counterexample:

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

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

    Quote Originally Posted by MachinePL1993 View Post
    $\displaystyle A_{1} $ and $\displaystyle B_{1} $ have the same cardinality, so do $\displaystyle A_{2} $ and $\displaystyle B_{2} $
    I need to check if the following sets also have the same cardinality:
    1)$\displaystyle A_{1} \times A_{2} $ and $\displaystyle 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 $\displaystyle f:A_{1} \rightarrow B_{1}$ and $\displaystyle f:A_{2} \rightarrow B_{2}$.

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

    Counterexample:

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

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

    Counterexample:

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

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

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

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

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