The cardinalities

Question

I have some difficulties to resolve a problem.

Could you explain me why this sets:

$[a,b]=[c,d]$,where $a,b,c,d \in\mathbb{R}$ with $a<b$ and $c<d$

have the same cardinalities?

thanks

Answer 1

Outline: We can find an explicit one to one correspondence $f$ between the two intervals. One way to find such a thing goes as follows. We try to find a function $f(x)=px+q$ which works. So we want $pa+q=c$ and $pb+q=d$. We can now solve this system of two linear equations for $p$ and $q$.

Answer 2

HINT: There is an easy bijection from $[a,b]$ to $[0,1]$ for all $a < b \in \mathbb{R}$

$$f(x) = \dfrac{x-a}{b-a}$$