Injective mapping I am reading about manifolds in the larger context of Cosmology. I came across the definition of injective mapping. The writer defines injective mapping as one in which one element of M maps to no more than one element of N. He further points out that $\phi(x)=e^x$ is an injective mapping. I can see how one element of $x$ maps to no more than one element of $\phi(x)$ but how it can it be less than one? I assuming that the definition of injective mapping allows the condition where one element of M does not map to any element of N. 

That is not the definition of an injective map. In order to get a map one needs that each element is only sent to one element. A map $f  \colon M \rightarrow N$ is injective if every element of $N$ has at most one preimage. Consider for example:

The element $3$ is ”hit“ more than one times by our map. Therefore the map I drew is not injective. Injective maps are maps such that each element is ”hit“ at most one times under the map.
