Show that $(0,1)$ and $(0,1]$ have the same cardinality.

Answer

Define a function $f:(0,1) \rightarrow (0,1]$ as

$$f(x) = \begin{cases} 2x, & \text{$x = 1/2^n$ for $n=1,2,\cdots$} \\ x, & \text{otherwise} \end{cases}$$

(1) Injective: suppose $f(x_1) = f(x_2)$. If $f(x_1)=f(x_2)=1/2^n$ for some $n \in \{0,1,2,\cdots\}$, then $x_1=1/2^{n+1}=x_2$. Otherwise, $x_1=f(x_1)=f(x_2)=x_2$.

(2) Surjective: suppose $y \in (0,1]$. If $y = 1/2^n$ for some $n \in \{0,1,2,\cdots\}$, then $y = f(1/2^{n+1})$. Otherwise, $y = f(y)$.


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