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Revision as of 23:42, 24 June 2009
, 23:42, 24 June 2009→Factoring Functions: center figures
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 prsent purposes:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
| |
| Source X = {1, 2, 3, 4, 5} |
| | o o o o o |
| f | \ | / \ / |
| | \|/ \ / |
| v o o o o o o |
| Target Y = {A, B, C, D, E, F} |
| |
o---------------------------------------o
</pre>
</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%"
|
<pre>
<pre>
o---------------------------------------o
| |
| Source X = {1, 2, 3, 4, 5} |
| | o o o o o |
| g | \ | / \ / |
| v \|/ \ / |
| Middle M = { b , e } |
| | | | |
| h | | | |
| v o o o o o o |
| Target Y = {A, B, C, D, E, F} |
| |
o---------------------------------------o
</pre>
</pre>
|}
Writing functional compositions <math>f = g \circ h</math> "on the right", we have the following data about the situation:
Writing functional compositions <math>f = g \circ h</math> "on the right", we have the following data about the situation:
