3

I want to find a bijective function from $(\frac{1}{2},1]$ into $[0,1]$. So, What is a bijective function $f:(\frac{1}{2},1]\to[0,1]$?

alpha
  • 71

1 Answers1

2

I'll show you a bijection $f:(0,1]\to[0,1]$. I hope you can transform this into what you want (using any other bijection $(1/2,1]\to (0,1]$).

It's actually pretty straightforward. Let $f(1)=0$, and $f(1/n)=1/(n-1)$ when $n\geq 1$ is an integer. This means that: $$f(1) = 0$$ $$f(1/2) = 1$$ $$f(1/3) = 1/2$$ $$f(1/4) = 1/3$$ $$\vdots$$ Well, now we have a bijection from $\left\{1/n : n\in\mathbb N\right\}$ to $\left\{1/n : n\in\mathbb N\right\}\cup\left\{0\right\}$. Now, we only need to define $f(x)=x$ when $x\in(0,1]$ is not of the form $1/n$ for any $n$.

fonini
  • 2,728