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I need to prove that $[-4,0)\sim(0,12)$ using the Cantor-Bernstein theorem and also by constructing a bijection.

From what I understood, in order to prove it with the Cantor theorem, I need to construct an injection between the two. From looking at one example I found, I saw that they did it by using a function.

Example was, from $(0,1)$ to $(0,1]$, $f(x) = x$, $x \in (0,1)$, because $(0,1) \subset (0,1]$

I tried to apply the same logic to my question, but got stuck..

I'm very new to this topic and I don't have a clue about where to start.

epiphany
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  • For an injection from the first set to the second, how about $f(x) = x+k$ for a constant $k$. What value of $k$ would work? Going the other way, you'll have to shrink the interval: try $g(x) = mx+l$ for some $m$ and $l$. Since the length of the interval $(0,12)$ is three times bigger than $[-4,0)$, you would want $m$ to be no bigger than $1/3$. – Théophile Dec 05 '20 at 16:41
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    thank you! I edited it! – epiphany Dec 05 '20 at 17:04

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Let's firstly consider $y=f(x)=3x+12$ which maps $(-4,0)$ to $(0,12)$. We can take some countable set from $(-4,0)$, for example $\left\{-\frac{1}{n}\right\}=\left\{x_n\right\}$ for $n\in \mathbb{N}$. By function $f$ these set goes to set $\left\{-\frac{3}{n}+12\right\}=\left\{y_n\right\}$ as $x_n \to y_n$. If now we consider set $\left\{4, x_n\right\}$, then is possible to define bijection $4 \to y_1, x_n \to y_{n+1}$.

zkutch
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