I need to find a bijection between the reals and (−∞, 0) however I'm struggling to do so. I'm trying to prove that these two have the same cardinality.

Also need to prove that the Reals and the interval (−∞, 0] have the same cardinality but I'm not quite sure on this one either.

For the first one, I tried using f(x)=e^x after seeing a previous question posted here but then soon realized that would not work because of the interval being from negative infinity to zero.

Thanks in advance to anyone who is able to help me.


The bijection can be realized by the negative of the exponential function: $$ f:\mathbb R\to (-\infty, 0)\quad x\mapsto - e^{x} $$


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