Hi guys I am having trouble understanding cardinality. I am given this practice question.
1) Use Cantor-Schroder-Bernstein Theorem to prove that the intervals $(0,1)$ and $[0,1]$ have the same cardinality?
My attempt :
I know the CSB theorem is
Let $A, B$ be sets and if $f: A \rightarrow B$ , and $g: B \rightarrow A$ are both injections, then there exists a bijection from $A$ to $B$
I have to show that $|(0,1)| = |[0,1]| $ so we need $f: (0,1) \rightarrow [0,1]$ and we can set $f(x) = x$ and also we need $g: [0,1] \rightarrow (0,1)$ and we can let $g(x) = \frac{1}{2}x + \frac{1}{4}$
but after I get stuck I dont know what to do i know have to set these equations up but get confused after.
Please help out any help or hints will be greatly appreciated
Thank you
You’re done as soon as you verify that the maps $f$ and $g$ are injections, that the range of $f$ is a subset of $[0,1]$, and that the range of $g$ is a subset of $(0,1)$: at that point you have an injection $f:(0,1)\to[0,1]$ and an injection $g:[0,1]\to(0,1)$, and the CSB theorem tells you outright that there exists a bijection $h:[0,1]\to(0,1)$ and hence that $\left|[0,1]\right|=\left|(0,1)\right|$.