Can anyone help me solve this?
Show that the interval (0,1] is equinumerous to the interval (0,1) by giving an example of a bijection from (0,1] to (0,1).
Thanks.
Let $\displaystyle A=(0,1]\subseteq\mathbb{R}$ and $\displaystyle B=(0,1)\subseteq\mathbb{R}$ and define $\displaystyle f:A\to B$ as follows:
$\displaystyle f(x)=\begin{cases}\frac{1}{2^{n+1}}\quad \exists\ n \in \mathbb{Z}: x=\frac{1}{2^n}\\ x\quad\ \ \not \exists\ n \in \mathbb{Z}: x=\frac{1}{2^n} \end{cases}$
This is very similar to the question in this thread.
I don't understand the point of having to exhibit the bijection. Clearly one exists since the countable union of countable sets is countable, and using this (decompose your set into the disjoint union of a countable set and an uncountable set) one can prove that if $\displaystyle E$ is uncountable then $\displaystyle E\cup N\simeq E$ for any $\displaystyle N\simeq\mathbb{N}$