I want to prove that any open interval $(a,b)$ has the same cardinality of the real numbers: $|(a,b)| = |\Bbb R|$.
Do I have to find an function to prove it? Or is there a theorem to prove it easier? or any idea?
[Math] $|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval
elementary-set-theory
Best Answer
The function $y = \tan(x)$ is bijective on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. We would like to shift/stretch it so that it's bijective on $(a,b)$. The period should be $b-a$, so we would at least have $y = \tan\left(\frac{\pi}{b-a}x \right)$. Now translate it.