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, 03:20, 14 April 2009→Commentary Note 11.10
In the case of a 2-adic relation ''F'' ⊆ ''X'' × ''Y'' that has the qualifications of a function ''f'' : ''X'' → ''Y'', there are a number of further differentia that arise:
In the case of a 2-adic relation ''F'' ⊆ ''X'' × ''Y'' that has the qualifications of a function ''f'' : ''X'' → ''Y'', there are a number of further differentia that arise:
:{| cellpadding="4"
{| cellpadding="4"
| ''f'' is "surjective"
| ''f'' is "surjective"
| iff
| iff
| iff
| iff
| ''f'' is 1-regular at ''Y''.
| ''f'' is 1-regular at ''Y''.
|}
|}
For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective.
For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective.
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
</pre>
|}
A cheap way of getting a surjective function out of any function is to reset its codomain to its range. For example, the range of the function ''f'' above is ''Y''′ = {0, 2, 5, 6, 7, 8, 9}. Thus, if we form a new function ''g'' : ''X'' → ''Y''′ that looks just like ''f'' on the domain ''X'' but is assigned the codomain ''Y''′, then ''g'' is surjective, and is described as mapping "onto" ''Y''′.
A cheap way of getting a surjective function out of any function is to reset its codomain to its range. For example, the range of the function ''f'' above is ''Y''′ = {0, 2, 5, 6, 7, 8, 9}. Thus, if we form a new function ''g'' : ''X'' → ''Y''′ that looks just like ''f'' on the domain ''X'' but is assigned the codomain ''Y''′, then ''g'' is surjective, and is described as mapping "onto" ''Y''′.
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 2 5 6 7 8 9
0 2 5 6 7 8 9
</pre>
</pre>
|}
The function ''h'' : ''Y''′ → ''Y'' is injective.
The function ''h'' : ''Y''′ → ''Y'' is injective.
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
0 2 5 6 7 8 9
0 2 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
</pre>
|}
The function ''m'' : ''X'' → ''Y'' is bijective.
The function ''m'' : ''X'' → ''Y'' is bijective.
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
</pre>
|}
===Commentary Note 11.11===
===Commentary Note 11.11===
