Use the Schröder-Bernstein theorem to show that there is a bijection between two intervals $[0,1]\subseteq \Bbb R$ and $[1,\infty)\subseteq \Bbb R$, thus they have the same cardinality.
What about the sets $(0,5)$ and $(10,20)$? Is there a bijection between them? (Don't need the Schröder-Bernstein theorem here). Similarly, consider $[0,\infty)$ and $[1,\infty)$.
Hello, this is a question from my practice final. Can anyone explain how to answer this question? As I understand, the theorem allows you to find a one-to-one function between two intervals to show that they have same cardinality, but I don't know how to apply this. For example, for the first question, is it simple as a function like $f(x) = 1/x$ or is there something more than that?
Best Answer
The Cantor-Bernstein theorem is probably one of the most useful and easily applied theorems in set theory.
That is all. Under the axiom of choice we can replace the injections with surjections, or so, but still constructing injections is often simple enough, especially at the level of this question.
So to use the theorem you really just need to come up with two injections $f\colon[0,1]\to[1,\infty)$ and $g\colon[1,\infty)\to[0,1]$. This will be sufficient to conclude that there is a bijection between these two subsets of $\Bbb R$.