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

[CubicFlunky77]

This is the first 'problem' in my Linear Algebra/Geometry textbook. 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: \mathbb R^+ \leftrightarrow \mathbb Q?

What I've done:

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


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


\forall (ε \in \mathbb R^+, c \in \mathbb Q) \in \mathbb K \exists (ε \cap c) \in (\mathbb R^+ \bigcap \mathbb Q) |\mathbb R^+ \leftrightarrow \mathbb Q

 

[Bacle2]

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)

 

[CubicFlunky77]

I get it, thanks!