Show that $\mathbb{R}$ and the open interval $(-1, 1)$ have the same cardinality.

Question

I am a little confused about using functions to show that two sets of intervals have the same cardinality.

I believe that if we can find a bijective function $f$ such that $f: \mathbb{R} \to (-1, 1)$, then they have the same cardinality? For example,

$$ f(x) = \tanh x $$

However, can we also say that if we can find a bijective function $f$ such that $f: (-1, 1) \to \mathbb{R}$, then they also have the same cardinality? If yes, can you provide an example?

Answer 1

Consider the function $f: \mathbf{R} \rightarrow (-1, 1)$ defined by $$ f(x) = {x \over 1 + | x |} $$

Clearly, $f$ is well-defined.

It is easy to verify that $f$ is a bijection.

In fact, the inverse of $f$ is obtained as $g: (-1, 1) \rightarrow \mathbf{R}$ defined by $$ g(y) = f^{-1}(y) = {y \over 1 - |y|} $$

Hence, it follows that $\mathbf{R}$ and $(-1, 1)$ have the same cardinality.