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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 20:35, 7 January 2013
, 20:35, 7 January 2013→6.34. Set-Theoretic Constructions: markup
<p>The larger and longer-term index sets, typically having the form <math>J \subseteq \mathbb{N} = \{ 1, 2, 3, \ldots \},\!</math> are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.</p></li></ol>
<p>The larger and longer-term index sets, typically having the form <math>J \subseteq \mathbb{N} = \{ 1, 2, 3, \ldots \},\!</math> are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.</p></li></ol>
'''Definition.''' An ''indicated set'' <math>j \widehat{~} S\!</math> is an ordered pair <math>j \widehat{~} S = (j, S),\!</math> where <math>j \in J\!</math> is the indicator of the set and <math>S\!</math> is the set indicated.
'''Definition.''' An ''indicated set'' <math>j \widehat{~} S\!</math> is an ordered pair <math>j \widehat{~} S = (j, S),\!</math> where <math>j \in J\!</math> is the ''indicator'' of the set and <math>S\!</math> is the set indicated.
'''Definition.''' An ''indited set'' <math>j \widehat{~} S\!</math> extends the incidental and extraneous indication of a set into a constant indictment of its entire membership.
Definition. An "indited set" j^S extends the incidental and extraneous indication of a set into a constant indictment of its entire membership.
j^S = j^{j^s : s C S} = j^{<j, s> : s C S} = <j, {j} x S>.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
j \widehat{~} S
& = & j \widehat{~} \{ j \widehat{~} s : s \in S \}
\\[4pt]
& = & j \widehat{~} \{ (j, s) : s \in S \}
\\[4pt]
& = & (j, \{ j \} \times S)
\end{array}</math>
|}
<pre>
Notice the difference between these notions and the more familiar concepts of an "indexed set", "numbered set", and "enumerated set". In each of these cases the construct that results is one where each element has a distinctive index attached to it. In contrast, the above indications and indictments attach to the set S as a whole, and respectively to each element of it, the same index number j.
Notice the difference between these notions and the more familiar concepts of an "indexed set", "numbered set", and "enumerated set". In each of these cases the construct that results is one where each element has a distinctive index attached to it. In contrast, the above indications and indictments attach to the set S as a whole, and respectively to each element of it, the same index number j.