Changes

Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').

Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').

<pre>

<blockquote>

| If P is a pre-function P : X ~> Y that happens to be total at X, then P

<p>If ''P'' is a pre-function ''P''&nbsp;:&nbsp;''X''&nbsp;~>&nbsp;''Y'' that happens to be total at ''X'', then ''P'' is known as a "function" from ''X'' to ''Y'', typically indicated as ''P''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y''.</p>

| is known as a "function" from X to Y, typically indicated as P : X -> Y.

 

<p>To say that a relation ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' is totally tubular at ''X'' is to say that it is 1-regular at ''X''.  Thus, we may formalize the following definitions:</p>

| To say that a relation P c X x Y is totally tubular at X is to say that

 

| it is 1-regular at X.  Thus, we may formalize the following definitions:

:{| cellpadding="4"

|

| ''P'' is a "function" ''p'' : ''X'' &rarr; ''Y''

| P is a "function" p : X -> Y   iff   P is 1-regular at X.

| iff

|

| ''P'' is 1-regular at ''X''.

| P is a "function" p : X <- Y   iff   P is 1-regular at Y.

|-

| ''P'' is a "function" ''p'' : ''X'' &larr; ''Y''

| iff

| ''P'' is 1-regular at ''Y''.

For example, let X = Y = {0, ..., 9} and let F c X x Y be

For example, let ''X'' = ''Y'' = {0,&nbsp;&hellip;,&nbsp;9} and let ''F''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' be the 2-adic relation that is depicted in the bigraph below:

the 2-adic relation that is depicted in the bigraph below:

0  1  2  3  4  5  6  7  8  9

0  1  2  3  4  5  6  7  8  9

o  o  o  o  o  o  o  o  o  o  X

o  o  o  o  o  o  o  o  o  o  X

o  o  o  o  o  o  o  o  o  o  Y

o  o  o  o  o  o  o  o  o  o  Y

0  1  2  3  4  5  6  7  8  9

0  1  2  3  4  5  6  7  8  9

We observe that F is a function at Y,

We observe that ''F'' is a function at ''Y'', and we record this fact in either of the manners ''F''&nbsp;:&nbsp;''X''&nbsp;&larr;&nbsp;''Y'' or ''F''&nbsp;:&nbsp;''Y''&nbsp;&rarr;&nbsp;''X''.

and we record this fact in either of

the manners F : X <- Y or F : Y -> X.

===Commentary Note 11.10===

===Commentary Note 11.10===