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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 20:22, 26 November 2007
, 20:22, 26 November 2007→Commentary Note 11.7: markup
We come now to the very special cases of 2-adic relations that are known as functions. It will serve a dual purpose on behalf of the exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material.
We come now to the very special cases of 2-adic relations that are known as functions. It will serve a dual purpose on behalf of the exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material.
To begin, let's recall the definition of a local flag:
To begin, let's recall the definition of a ''local flag'':
<pre>
: ''L''<sub>''x''.''j''</sub> = { (''x''<sub>1</sub>, …, ''x''<sub>''j''</sub>, …, ''x''<sub>''k''</sub>) ∈ ''L'' : ''x''<sub>''j''</sub> = ''x'' }.
</pre>
In the light of these considerations, the local flags of a 2-adic relation L c X x Y may be formulated as follows:
In the case of a 2-adic relation ''L'' ⊆ ''X''<sub>1</sub> × ''X''<sub>2</sub> = ''X'' × ''Y'', we can reap the benefits of a radical simplification in the definitions of the local flags. Also in this case, we tend to denote ''L''<sub>''u''.1</sub> by "''L''<sub>''u''.''X''</sub>" and ''L''<sub>''v''.2</sub> by "''L''<sub>''v''.''Y''</sub>".
In the light of these considerations, the local flags of a 2-adic relation ''L'' ⊆ ''X'' × ''Y'' may be formulated as follows:
{| cellpadding="4"
| ''L''<sub>''u''.''X''</sub>
| =
| {(x, y) ∈ ''L'' : ''x'' = ''u''}
|-
|
| =
| the set of all ordered pairs in ''L'' incident with ''u'' in ''X''.
|-
| ''L''<sub>''v''.''Y''</sub>
| =
| {(''x'', ''y'') ∈ ''L'' : ''y'' = ''v''}
|-
|
| =
| the set of all ordered pairs in ''L'' incident with ''v'' in ''Y''.
|}
A sufficient illustration is supplied by the earlier example ''E''.
A sufficient illustration is supplied by the earlier example ''E''.
</pre>
</pre>
The local flag E_3@X is displayed here:
The local flag ''E''<sub>3.''X''</sub> is displayed here:
<pre>
<pre>
o o o o o o o o o o X
o o o o o o o o o o X
/|\
/|\
/ | \ E_3@X
/ | \ E_3.X
/ | \
/ | \
o o o o o o o o o o Y
o o o o o o o o o o Y
</pre>
</pre>
The local flag E_2@Y is displayed here:
The local flag ''E''<sub>2.''Y''</sub> is displayed here:
<pre>
<pre>
o o o o o o o o o o X
o o o o o o o o o o X
\ | /
\ | /
\ | / E_2@Y
\ | / E_2.Y
\|/
\|/
o o o o o o o o o o Y
o o o o o o o o o o Y