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Math Help - cardinals of finite sets

  1. #1
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    cardinals of finite sets

    Hello,


    I have the following question:
    Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.


    I find it hard to prove it because I can easily find a set A which is:


    A={1,2}
    B={1,2,3,4}
    C={1,2,3}


    C is a subset of B. C's cardinality is bigger than A's.


    There is an injective function from A to B, of course.


    where am I wrong?

    Thank you!
    Last edited by nerazzurri10; December 7th 2013 at 08:12 AM.
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  2. #2
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    Re: cardinals of finite sets

    A and B have the same cardinality if there is a 1-1 mapping of A onto B (bijective).

    If there is a 1-1 mapping of A onto a subset of B (the same cardinality), there exists a 1-1 mapping into B (injective).
    If there is an injective mapping into a subset of B, The mapping is onto that subset from A, and A and that subset have the same cardinaity.
    Last edited by Hartlw; December 7th 2013 at 09:29 AM. Reason: surjective should be bijective
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  3. #3
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    Re: cardinals of finite sets

    Quote Originally Posted by nerazzurri10 View Post
    Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.


    I find it hard to prove it because I can easily find a set A which is:


    A={1,2}
    B={1,2,3,4}
    C={1,2,3}


    C is a subset of B. C's cardinality is bigger than A's.


    There is an injective function from A to B, of course.


    where am I wrong?
    If there is an injective function from A to B, it means that A has the same cardinality as some subset of B, not every subset of B. So, C may have a greater cardinality than A's, but there is another subset of B that has the same cardinality as A's.
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  4. #4
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    Re: cardinals of finite sets

    Quote Originally Posted by nerazzurri10 View Post
    I have the following question:
    Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.
    Let's first agree on notation and definitions. By \|A\| I mean the cardinality of the set A. .
    Then \|A\|=\|B\|\text{ if and only if }A\overset{f} \leftrightarrow B, i.e. f is a bijection.

    To start suppose that there is an injection \phi : A\to B then clearly \phi(A)\subseteq B.
    Prove that A\overset{\phi} \leftrightarrow \phi(A).

    For the converse, assume that (\exists C\subseteq B)[A\overset{\rho} \leftrightarrow C] where \rho is a bijection.
    What is your injection?
    Last edited by Plato; December 7th 2013 at 09:01 AM.
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  5. #5
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    Re: cardinals of finite sets

    Nice example of using symbology to reword an existing proof (#2)
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