[SOLVED] Getting crazy with 2 cardinal numbers questions.

**aurora** · May 23rd 2009, 03:05 PM

Hey everybody, thanks for reading.
I started studying cardinal numbers not long ago, and very lately arithmetics of cardinal numbers.
I'm used to finding cardinal numbers using bijections, not with arithmetics, because I don't quite know how to use it.
The thing is, the excersices got really tricky and I don't seem to be able to find appropriate bijections.
So I'd appreciate help of any kind!
Here are the questions:
1) here's a set: A = {p(pi)/q(pi) |p(x),q(x) are in Q[x])}. In words: elements in A are all the ratios between two rational polynomials, in which instead of "x"you write "pi" (the irrational famous number!)

This is actually the minimal field that contains pi and Q'...
I need to find it's cardinal number.
2) here's a set: A = {all the converging sequences in R}.
Again, I need to find it's cardinality.

I really tried to solve these for hours...
I won't write "my attempt" just because I have no idea how to write alef here, and stuff, and it'll take forever (although it's my native tounge, hebrew, lol)...
So I really really appreciate it.
Thanks,
Tomer.

**NonCommAlg** · May 23rd 2009, 07:12 PM

Originally Posted by aurora

Hey everybody, thanks for reading.
I started studying cardinal numbers not long ago, and very lately arithmetics of cardinal numbers.
I'm used to finding cardinal numbers using bijections, not with arithmetics, because I don't quite know how to use it.
The thing is, the excersices got really tricky and I don't seem to be able to find appropriate bijections.
So I'd appreciate help of any kind!
Here are the questions:
1) here's a set: A = {p(pi)/q(pi) |p(x),q(x) are in Q[x])}. In words: elements in A are all the ratios between two rational polynomials, in which instead of "x"you write "pi" (the irrational famous number!)

This is actually the minimal field that contains pi and Q'...
I need to find it's cardinal number.
2) here's a set: A = {all the converging sequences in R}.
Again, I need to find it's cardinality.

I really tried to solve these for hours...
I won't write "my attempt" just because I have no idea how to write alef here, and stuff, and it'll take forever (although it's my native tounge, hebrew, lol)...
So I really really appreciate it.
Thanks,
Tomer.

i'll do the first question because it's algebra! haha

the standard notation for $A$ is $\mathbb{Q}(\pi).$ since $\pi$ is transcendental over $\mathbb{Q},$ we have $\mathbb{Q}(\pi) \cong \mathbb{Q}(x),$ where $\mathbb{Q}(x)$ is the field of rational functions. now we have two obvious injections $\mathbb{Q}[x] \to \mathbb{Q}(x)$

and $\mathbb{Q}(x) \to \mathbb{Q}[x] \times \mathbb{Q}[x],$ which gives us $|\mathbb{Q}(x)|=|\mathbb{Q}[x]|=|\mathbb{Q}|=\aleph_0,$ because for any infinite set $R$ we have $|R \times R|=|R|.$

a general fact is that for any integral domain $D$ we have $|D|=|Q(D)|,$ where $Q(D)$ is the quotient field (the field of fractions) of $D.$

the reason is that if $D$ is finite, then $D=Q(D)$ and if

$D$ is infinite then, as above, look at the injections $D \to Q(D)$ and $Q(D) \to D \times D.$

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