Changes

I will take this up first in the simplest possible setting, where it has to do with the special sorts of relations that are commonly called ''functions'', and after the basic idea is made as clear as possible in this easiest case I will deal with the notion of factorization as it affects more generic types of relations.

I will take this up first in the simplest possible setting, where it has to do with the special sorts of relations that are commonly called ''functions'', and after the basic idea is made as clear as possible in this easiest case I will deal with the notion of factorization as it affects more generic types of relations.

Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''.  Here is a picture of such function, <math>f : X \to Y,</math> as generic as it needs to be for our prsent purposes:

Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''.  Here is a picture of such function, <math>f : X \to Y,</math> as generic as it needs to be for our present purposes:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="8" style="text-align:center"

|

| [[Image:Factorization Function Example 1.jpg|500px]]

|-

o---------------------------------------o

| <math>\text{Figure 1.  Function}~ f : X \to Y</math>

|                                       |

|  Source X  =  {1, 2, 3, 4,    5}     |

|      f   |      \ | /    \  /      |

|  Target Y =  {p, q, r, s, t, u}    |

</pre>

|}

|}

It is a fact that any old function that you might pick "factors" into a functional composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:

It is a fact that any old function that you might pick factors into a functional composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="10" style="text-align:center; width:90%"