# Thread: [SOLVED] Getting crazy with 2 cardinal numbers questions.

1. ## [SOLVED] Getting crazy with 2 cardinal numbers questions.

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.

2. Originally Posted by aurora
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 $\displaystyle A$ is $\displaystyle \mathbb{Q}(\pi).$ since $\displaystyle \pi$ is transcendental over $\displaystyle \mathbb{Q},$ we have $\displaystyle \mathbb{Q}(\pi) \cong \mathbb{Q}(x),$ where $\displaystyle \mathbb{Q}(x)$ is the field of rational functions. now we have two obvious injections $\displaystyle \mathbb{Q}[x] \to \mathbb{Q}(x)$

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

a general fact is that for any integral domain $\displaystyle D$ we have $\displaystyle |D|=|Q(D)|,$ where $\displaystyle Q(D)$ is the quotient field (the field of fractions) of $\displaystyle D.$ the reason is that if $\displaystyle D$ is finite, then $\displaystyle D=Q(D)$ and if

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