Please help me prove the following equalities between set cardinalities by explicitly showing an appropriate mapping:
$$\left | (0,1) \right |= \left | (1,+\infty ) \right |$$
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The function $f(x)=1/x$ will suffice to form a bijection between $(0,1)$ and $(1, \infty)$. I'm not sure how much you know about comparing cardinalities, but the standard method is to form a bijective function between the two sets to be compared. Is it clear to you how this function does that?