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Math Help - [SOLVED] Getting crazy with 2 cardinal numbers questions.

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

    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.
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  2. #2
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    Quote Originally Posted by aurora View Post
    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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