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- One of the important things we will consider in this course is the idea of a function (most often from
- A map is
**one-one**(or*injective*) if two distinct elements of*A*never map to the same element of*B*.

That is,*f*:*A*→*B*is one-one if (*x*,*y*∈*A*)(*f*(*x*) =*f*(*y*) ⇒*x*=*y*)A map is

**onto**(or*surjective*) if every element of*B*has some element of*A*mapped to it.

That is,*f*:*A*→*B*is onto if (*y*∈*B*)(*x*∈*A*)(*f*(*x*) =*y*)A map which is both one-one and onto is called a

**one-one correspondence**(or*bijection*).The set

*A*is called the**domain**of the map*f*:*A*→*B*and the set*B*is sometimes called the**codomain**of*f*.The set of all

*b*∈*B*with*f*(*a*) =*b*for some*a*∈*A*is called the**image**(or*range*) of*f*.

You can see some of the early attempts.

As far as we are concerned a **map** or **function** *f* : *A*→ *B* is a method of assigning to every (*****) element *a* ∈ *A* a unique (*****) element *f*(*a*) ∈ *B*.

To do this rigorously (we will not, mostly) you can define a function *f* to be a *subset* of *A* × *B* (which looks a lot like the graph of the function) and include rules which ensure the two properties (*****) above hold.

**Definitions**

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