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Trying to show that rationals exist on the + real number line field K

  1. Jul 7, 2013 #1
    This is the first 'problem' in my Linear Algebra/Geometry text book. I just need to know if I am doing it correctly. Any hints? I also need to know if I am using correct notation/presentation.

    Question: [itex]\mathbb R^+ \leftrightarrow \mathbb Q[/itex]?

    What I've done:

    Suppose: [itex](ε_1,...,ε_n) \in \mathbb R^+ \rightarrow \mathbb K[/itex] and
    [itex](c_1,...,c_n) \in \mathbb Q \rightarrow \mathbb K[/itex]

    Assuming: [itex]\mathbb R^+ ⊂ \mathbb K[/itex] and [itex]\mathbb Q ⊂ \mathbb K[/itex] where [itex]\mathbb K[/itex] is a numerical/object field; we can say that

    [itex] \forall (ε \in \mathbb R^+, c \in \mathbb Q) \in \mathbb K \exists (ε \cap c) \in (\mathbb R^+ \bigcap \mathbb Q)[/itex] |[itex]\mathbb R^+ \leftrightarrow \mathbb Q[/itex]
  2. jcsd
  3. Jul 7, 2013 #2


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    Science Advisor

    I'm not sure I understood what you meant, but you can do this:

    i)1 is in ℝ , 1 as the identity, since ℝ is a field.

    ii) 1+1=2 is in ℝ , by closure of operations

    iii) (1+1)(-1)=1/2 is in ℝ , since every element in a field is invertible

    iv)..... Can you take it from here ( if this is what you meant)
  4. Jul 11, 2013 #3
    I get it, thanks!
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