surjection, injection and bijection clarification

[fullofquestions]

The way I see it if f: A -> B

surjection: EVERY element of B receives a map from A. Multiple elements of A can map to the same B.

injection: EVERY element of A maps to a unique element of B. Can there be B's left unmapped?

bijection (both surjective and injective): EVERY element of A maps to a unique element of B. There are no B's left unmapped, ie. # elements in A = # elements in B

Question1: Can the function that maps the empty set to the empty set be considered any of these?

Question 2
Many definitions of injective functions has been as follows and it leads me to think that in an injective function all elements of B receive a map. Is it true? I think not but I just want to make sure it is clear:

"If we have a function f : A -> B, such that every element of B is mapped onto by one and only one element of A the function is called a One-to one function or Injective function. "[/url]