Changes

We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:

We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:

: <math>X^2 = \{ (u, v) : u = v \}\ \cup\ \{ (u, v) : u \ne v \}</math>

: <math>\begin{matrix}

X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.

\end{matrix}</math>

In symbolic terms, this partition may be expressed as:

This partition may also be expressed in the follwing symbolic form:

: <math>\begin{matrix}

: ''X''² <math>\cong</math> Diag(''X'') + 2 * Comb(''X'', 2),

The separate terms of this formula are defined as follows:

where:

: <math>\begin{matrix}

: Diag(''X'') = {‹''x'', ''x''› : ''x'' &isin; ''X''},

: <math>\begin{matrix}

\tbinom{X}{k} & = & X\ \operatorname{choose}\ k & = & \{ k\!\mbox{-sets from}\ X \}.

\end{matrix}</math>

and where:

Thus we have:

: Comb(''X'', ''k'') = "''X'' choose ''k''" = {''k''-sets from ''X''},

: <math>\begin{matrix}

 

\tbinom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.

 

:  Comb(''X'', 2) = {{''u'', ''v''} : ''u'', ''v'' &isin; ''X''}.

We can now use the features in d<font face="lucida calligraphy">X</font> = {d''x''_''i''} = {d''x''₁,&nbsp;&hellip;,&nbsp;d''x''_''n''} to classify the paths of ('''B''' &rarr; ''X'') by way of the pairs in ''X''².  If ''X'' <math>\cong</math> '''B'''^''n'' then a path in ''X'' has the form ''q'' : ('''B''' &rarr; '''B'''^''n'') <math>\cong</math> '''B'''^''n'' &times; '''B'''^''n'' <math>\cong</math> '''B'''^2''n'' <math>\cong</math> ('''B'''²)^''n''.  Intuitively, we want to map this ('''B'''²)^''n'' onto ''D''^''n'' by mapping each component '''B'''² onto a copy of '''D'''.  But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.

We can now use the features in d<font face="lucida calligraphy">X</font> = {d''x''_''i''} = {d''x''₁,&nbsp;&hellip;,&nbsp;d''x''_''n''} to classify the paths of ('''B''' &rarr; ''X'') by way of the pairs in ''X''².  If ''X'' <math>\cong</math> '''B'''^''n'' then a path in ''X'' has the form ''q'' : ('''B''' &rarr; '''B'''^''n'') <math>\cong</math> '''B'''^''n'' &times; '''B'''^''n'' <math>\cong</math> '''B'''^2''n'' <math>\cong</math> ('''B'''²)^''n''.  Intuitively, we want to map this ('''B'''²)^''n'' onto ''D''^''n'' by mapping each component '''B'''² onto a copy of '''D'''.  But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.