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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 04:00, 13 April 2009
, 04:00, 13 April 2009→Commentary Note 11.8
Clearly, if any relation is <math>(\le c)\text{-regular}</math> on one of its domains <math>X_j\!</math> and also <math>(\ge c)\text{-regular}</math> on the same domain, then it must be <math>(= c)\text{-regular}\!</math> on that domain, in effect, <math>c\text{-regular}\!</math> at <math>j.\!</math>
Clearly, if any relation is <math>(\le c)\text{-regular}</math> on one of its domains <math>X_j\!</math> and also <math>(\ge c)\text{-regular}</math> on the same domain, then it must be <math>(= c)\text{-regular}\!</math> on that domain, in effect, <math>c\text{-regular}\!</math> at <math>j.\!</math>
For example, let ''G'' = {''r'', ''s'', ''t''} and ''H'' = {1, …, 9}, and consider the 2-adic relation ''F'' ⊆ ''G'' × ''H'' that is bigraphed here:
For example, let <math>G = \{ r, s, t \}\!</math> and <math>H = \{ 1, \ldots, 9 \},\!</math> and consider the 2-adic relation <math>F \subseteq G \times H</math> that is bigraphed here:
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
r s t
r s t
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
</pre>
</pre>
|}
We observe that ''F'' is 3-regular at ''G'' and 1-regular at ''H''.
We observe that <math>F\!</math> is 3-regular at <math>G\!</math> and 1-regular at <math>H.\!</math>
===Commentary Note 11.9===
===Commentary Note 11.9===