R or R*: Finding the Subsets and Containing Coordinates

[gotjrgkr]

Let R be a set of real numbers derived from rational numbers and R* be a set consisting of all ordered pairs of the form (x,0) where x is contained in R.
Then R* can be identified with R.

I'd like to ask you two questions.

1. Definition of definite integral of complex valued function of real variable.
Let f be a complex valued function of real variable on [a,b] into C where C is the set of complex numbers ; C is the set of all ordered pairs of the form (x,y) with x,y contained in R.
Let f(t)= f_1(t) + i * f_2(t)
Then the definite integral of f is defined as follows
; integral of f(t) from a to b = integral of f_1(t) from a to b + i * integral of f_2(t) from a to b.
What I want to know is that which set between R and R* has [a,b] as its subset.

2. Definition of k-dimensional Euclidean space.
Let J(k) be the k-dimensional Euclidean space.
Then , if x is contained in the Euclidean space, x= (x_1,x_2,...,x_k).

I'd like to know that which set between R and R* contains a coordinate x_i of x where i runs from 1 to k.

 

[HallsofIvy]

I have absolutely know clue what you mean by "which set between R and R*". R and R* are distinctly different sets and there is no set "between" them.

 

[gotjrgkr]

I have absolutely know clue what you mean by "which set between R and R*". R and R* are distinctly different sets and there is no set "between" them.

I'm sorry to confuse you.
I'm living a country where English is not a first language.(I'm not good at speaking english)
So, I hope you to try to understand my mistakes.
Of course, there is no set "between" them.
i'll try to explain it by another way.

In question 1,
let A be a statement such that [a,b] is a subset of R,
and let B be a statement such that [a,b] is a subset of R*.
In these two cases A,B, which case is true? is it A? or is it B?

In question 2,
let A' be a statement such that x_i is an element of R,
and let B' be a statement such that x_i is an element of R*.
In these two cases A',B', which case is true? which statement is true?
is it A'? or is it B'?

I really hope this additional explanation help you to understand my question.
Thanks.