Help with this question would be greatly appreciated. At the moment, I cannot find an f(x) to map these sets.
Also I'm just wondering to solve this problem, since '[]' brackets are used, do the numbers 0 and 1 have to be mapped to from the set (0, ∞), because that really makes the question quite tough, otherwise something like 1/Squareroot(x+1)
p.s. can someone link me the MathJax formatting, cant find it
edit: (0, ∞) is not the reals as it does not go to negative
Thanks!
If you'd asked for a proof $(0,\,\infty)$ has the same cardinality as $(0,\,1)$, the trick would be easy: specify a bijection as $x\mapsto\frac{1}{\sqrt{1+x}}$ (or whatever choice you prefer; personally, I wouldn't bother with the square-rooting). Your actual question is about $[0,\,1]$, so what do we do? Well, on the one hand we now know $(0,\,\infty)$ can be injected into $[0,\,1]$. On the other hand, the reverse injection is easy (e.g. $x\mapsto x+1$); now finish with the Schröder–Bernstein theorem, which states if two sets can be injected into each other there's a bijection between them. This theorem is used so often in same-cardinality proofs people don't even mention its usage explicitly. Best of all, its proof is constructive, i.e. it tells you how to form the bijection, if you really want one.