Basic definitions dealing with functions
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Let A and B be nonempty sets.

Then a *function* f: A -> B associates to each a\in A a particular point f(a) in B.  
a function is sometimes defined as a relation f in which  
there for every a in A there is one and only one (a,b)\in f.
A *map* is another word for a function.

The value f(a) is sometimes called the *image* of a.
If f(a)=b, we call a a *preimage* of b. 

The set A is called the *domain* of f.
This is denoted Domain(f).

The set of all points b \in B such that b=f(a) for some a\in A 
is called the *range* of f.
The is denoted Range(f).

The set B doesn't have a common name.  (Sometimes people say range
in reference to B, but I'll try not do do that.)   Sometimes people
call it the "target" of f. 

The function f is *onto*, or *surjective*, 
if B = Range(f): that is, every b in B has some preimage.

The function f is *one-to-one*, or *injective*, if f(a)=f(a') implies a=a'.
In other words, a function f is one-to-one if no two points map to the
same image.

A function is *bijective* (it is a *bijection*) if
it is both one-to-one and onto.