Changes

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 <math>S\!</math> as a whole, and respectively to each element of it, the same index number <math>j.\!</math>

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 <math>S\!</math> as a whole, and respectively to each element of it, the same index number <math>j.\!</math>

'''Definition.''' An ''indexed set'' <math>(S, L)\!</math> is constructed from two components:  its ''underlying set'' <math>S\!</math> and its ''indexing relation'' <math>L : S \to \mathbb{N},\!</math> where <math>L\!</math> is total at <math>S\!</math> and tubular at <math>\mathbb{N}.\!</math> It is defined as follows:

Definition.  An "indexed set" <S, L> is constructed from two components:  its "underlying set" S and its "indexing relation" L : S > N, where L is total at S and tubular at N.  It is defined as follows:

<S, L>  = { {s} x L(s) : s C S } = {<s, j> : s C S, j C L(s)}.

| <math>(S, L) = \{ \{ s \} \times L(s) : s \in S \} = \{ (s, j) : s \in S, j \in L(s) \}.\!</math>

L assigns a unique set of "local habitations" L(s) to each element s in the underlying set S.

<math>L\!</math> assigns a unique set of &ldquo;local habitations&rdquo; <math>L(s)\!</math> to each element <math>s\!</math> in the underlying set <math>S.\!</math>

Definition.  A "numbered set" <S, f>, based on the set S and the injective function f : S  > N, is defined as follows.  ???

Definition.  A "numbered set" <S, f>, based on the set S and the injective function f : S  > N, is defined as follows.  ???