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:
$(a,b)$
$(a,b]$
$[a, b)$
$[a, b]$
$(a, \infty)$
$[a, \infty)$
$(-\infty, b)$
$(-\infty, b]$
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.
Best Answer
HINT: Recall that if $X\subseteq Y\subseteq Z$ and $X$ and $Z$ are equinumerous, then $Y$ and $Z$ are equinumerous.