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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 03:36, 5 December 2007
, 03:36, 5 December 2007→Commentary Note 11.10: markup
===Commentary Note 11.10===
===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 c X x Y that has
the qualifications of a function f : X -> Y, there
are a number of further differentia that arise:
| f is "surjective" iff f is total at Y.
<blockquote>
|
{| cellpadding="4"
| f is "injective" iff f is tubular at Y.
| ''f'' is "surjective"
|
| iff
| f is "bijective" iff f is 1-regular at Y.
| ''f'' is total at ''Y''.
|-
| ''f'' is "injective"
| iff
| ''f'' is tubular at ''Y''.
|-
| ''f'' is "bijective"
| iff
| ''f'' is 1-regular at ''Y''.
|}
</blockquote>
For example, or more precisely, contra example,
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.
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.
<pre>
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
</pre>
A cheap way of getting a surjective function out of any function
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''′.
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'.
<pre>
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 Y'
o o o o o o o Y'
0 2 5 6 7 8 9
0 2 5 6 7 8 9
</pre>
The function h : Y' -> Y is injective.
The function ''h'' : ''Y''′ → ''Y'' is injective.
<pre>
0 2 5 6 7 8 9
0 2 5 6 7 8 9
o o o o o o o Y'
o o o o o o o Y'
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
</pre>
The function m : X -> Y is bijective.
The function ''m'' : ''X'' → ''Y'' is bijective.
<pre>
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