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Math Help - question about cardinality of empty set

  1. #1
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    question about cardinality of empty set

    Hi

    I am doing the chapter "Equinemerous sets" from Velleman's "How to prove it" and
    I have some doubts. At one point he says that "for each natural number n , let
    I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}. A set A
    is called \mathit{finite} if there is a natural number n such that
    I_n \sim A . Otherwise A is \mathit{infinite}. "

    Further down he says that "it makes sense to define the number of elements of
    a finite set A to be the unique n such that I_n \sim A. This number
    is also sometimes called the cardinality of A and its denoted
    \lvert A \rvert. Note that according to this definition,
    \varnothing is finite and \lvert \varnothing \rvert =0."

    So that will mean that we will need to choose n=0 for an empty set, so that
    I_0 \sim \varnothing. Now according to the author's defnition of
    I_n , I_0=\varnothing. So

    I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing

    which is true since for any set A, we have A \sim A.

    Do you think its correct understanding ?

    Thanks
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  2. #2
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    Re: question about cardinality of empty set

    Quote Originally Posted by issacnewton View Post
    At one point he says that "for each natural number n , let
    I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}. A set A
    is called \mathit{finite} if there is a natural number n such that
    I_n \sim A . Otherwise A is \mathit{infinite}. "
    Further down he says that "it makes sense to define the number of elements of
    a finite set A to be the unique n such that I_n \sim A. This number
    is also sometimes called the cardinality of A and its denoted
    \lvert A \rvert. Note that according to this definition,
    \varnothing is finite and \lvert \varnothing \rvert =0."

    So that will mean that we will need to choose n=0 for an empty set, so that
    I_0 \sim \varnothing. Now according to the author's definition of
    I_n , I_0=\varnothing. So I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing which is true since for any set A, we have A \sim A.
    Do you think its correct understanding ?
    That is consistent with his definition because S_0 must be \emptyset.
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  3. #3
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    Re: question about cardinality of empty set

    What is S_0 ?
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    Re: question about cardinality of empty set

    Quote Originally Posted by issacnewton View Post
    What is S_0 ?
    Well look at Velleman's own definition: S_n=\{i\in\mathbb{Z}^+:i\le n\}.
    Now what positive integer is less than or equal to zero?

    So answer your own question.

    Note that Velleman did not specify the nature of n in that definition.
    So S_{0.5}=S_0
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  5. #5
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    Re: question about cardinality of empty set

    Oh you meant I_0. Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...

    Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving
    the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
    it makes sense to give definitions as rigorous as possible
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    Re: question about cardinality of empty set

    Quote Originally Posted by issacnewton View Post
    Oh you meant I_0. Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...
    Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
    it makes sense to give definitions as rigorous as possible
    This may surprise you but I have never seen that textbook.
    But if you look at Velleman's pedigree you see that one of his advisers was Mary Ellen Rudin (Walter Rudin's wife) and an R L Moore's PhD student. That should at once tell you that her students should be a stickler for precise definitions. There you have some of the best mathematicians of the last century . So if I were you I would rethink that remark.
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  7. #7
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    Re: question about cardinality of empty set

    Well, thanks for the information. Not being from US, didn't know that.

    Its strange that even the real analysis books I have seen, don't use such precise
    definitions. So that style is not followed everywhere. I was checking Moore's wikipedia page and there is mention of Moore's method. May be this style comes from there....
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