1. 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

2. Re: question about cardinality of empty set

Originally Posted by issacnewton
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 $\displaystyle \emptyset$.

3. Re: question about cardinality of empty set

What is $\displaystyle S_0$ ?

4. Re: question about cardinality of empty set

Originally Posted by issacnewton
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?

Note that Velleman did not specify the nature of n in that definition.
So $\displaystyle S_{0.5}=S_0$

5. 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

6. Re: question about cardinality of empty set

Originally Posted by issacnewton
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.

7. 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....