elementary set theory - Proving intervals are equinumerous to $mathbb R$

Let $a$, $b$ elements of $\mathbb R$ with $a < b$. By combining the results of the past exercises and examples, show that each of the following intervals are equinumerous to the set $\mathbb R$ of all real numbers:

1. $(a,b)$




2. $(a,b]$




3. $[a, b)$




4. $[a, b]$




5. $(a, \infty)$




6. $[a, \infty)$





7. $(-\infty, b)$




8. $(-\infty, b]$

Do I need to find a bijection or what, from each interval to $(-\infty, \infty)$? We've shown in class that $(a,b)$ is equinumerous to $(0,1)$, and that $(0,1)$ is equinumerous to $\mathbb R$. We have gone over a proof that $[a, b]$ is equinumerous to $[0,1]$. We've also covered that $[a, b]$ and $(a,b)$ have the same cardinality.

Doing 8 proofs of intervals with a function $f$ being a bijection to $\mathbb R$ seems really tedious, so I'm asking if there's another way around that?

And if there isn't, I'd like some help on the proofs.