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Thread: 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
    $\displaystyle I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$. A set $\displaystyle A$
    is called $\displaystyle \mathit{finite}$ if there is a natural number n such that
    $\displaystyle I_n \sim A$ . Otherwise A is $\displaystyle \mathit{infinite}$. "

    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 $\displaystyle I_n \sim A$. This number
    is also sometimes called the cardinality of A and its denoted
    $\displaystyle \lvert A \rvert$. Note that according to this definition,
    $\displaystyle \varnothing$ is finite and $\displaystyle \lvert \varnothing \rvert =0$."

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

    $\displaystyle I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$

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

    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
    $\displaystyle I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$. A set $\displaystyle A$
    is called $\displaystyle \mathit{finite}$ if there is a natural number n such that
    $\displaystyle I_n \sim A$ . Otherwise A is $\displaystyle \mathit{infinite}$. "
    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 $\displaystyle I_n \sim A$. This number
    is also sometimes called the cardinality of A and its denoted
    $\displaystyle \lvert A \rvert$. Note that according to this definition,
    $\displaystyle \varnothing$ is finite and $\displaystyle \lvert \varnothing \rvert =0$."

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

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

    Quote Originally Posted by issacnewton View Post
    What is $\displaystyle S_0$ ?
    Well look at Velleman's own definition: $\displaystyle S_n=\{i\in\mathbb{Z}^+:i\le n\}$.
    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 $\displaystyle S_{0.5}=S_0$
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  5. #5
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    Re: question about cardinality of empty set

    Oh you meant $\displaystyle I_0$. 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 $\displaystyle I_0$. 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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