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

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In summary, the question is asking if the sets of positive real numbers and rational numbers are equivalent. The individual has provided some steps towards solving the problem, including defining the sets and using closure of operations and invertibility. They are unsure if they have understood the question correctly and ask for further guidance.
  • #1
CubicFlunky77
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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: [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]
 
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  • #2
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)
 
  • #3
I get it, thanks!
 

What are rationals and how do they relate to the real number line field K?

Rationals are numbers that can be expressed as a ratio of two integers, such as fractions. They are a subset of the real number line field K, which also includes irrational numbers like pi and square root of 2.

How can we prove that rationals exist on the real number line field K?

One way to prove this is by using the Dedekind cut method, which involves dividing the real number line into two sets where all the numbers in one set are less than those in the other set. This can be used to show that there is no gap on the real number line where rationals cannot exist.

Can you provide an example of how to represent a rational number on the real number line field K?

Yes, for example, the rational number 1/2 can be represented on the real number line field K as the point located exactly halfway between 0 and 1 on the number line.

What is the significance of proving that rationals exist on the real number line field K?

Proving this shows that the real number line is a complete and continuous set of numbers, and that there are no gaps or missing numbers. It also allows for the representation and calculation of fractions and other rational numbers on the real number line.

Are there any other methods or proofs for demonstrating the existence of rationals on the real number line field K?

Yes, there are other methods such as using continued fractions to represent rationals on the real number line, or using the concept of equivalence classes to show that every rational number has a unique representation on the real number line. However, the Dedekind cut method is the most commonly used proof.

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