Basic definitions dealing with functions ---------------------------------------- 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.