Show that any interval [A, B] of the number axis is equivalent to any other interval [C, D].

Question

I am attempting to get my head around intervals, particularly the title question as described in What is Mathematics? (Courant & Stewart).

I think I am probably misunderstanding the meaning of equivalence in this context. My interpretation of the text is that intervals contain an infinite set of real numbers of increasing accuracy between two bounds, so how is it that two intervals [A, B] and [C, D] could be equivalent? Surely, by Cantor's diagonal argument, sets of real numbers contain irrational numbers that cannot be counted.

As you can't create a bi-unique relationship utilising the set of real numbers, how could you prove that two intervals have the same cardinality? Intuitively to me I can't determine how there is any way to pair the irrational numbers together.

I have also considered that my understanding of intervals is incorrect, do they actually contain irrational numbers? Perhaps they provide a means to describe irrational numbers as opposed to actually encapsulating them? Thanks in advance.

Answer 1

Why if you just need a bijection from [a,b] to [c,d] here it is :

$$ \frac{c(b-x) + (x-a)d}{(b-a)}.$$ Check its a bijection.

Answer 2

Definition of equal Cardinality: Two sets X and Y are said to be of equal cardinality if there is a bijection from X $\rightarrow$ Y.

Now getting a function which takes one interval to other is quite easy to find.