Changes

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|+ <math>\text{Table 43.}~~\text{Composite and Compiled Order Relations}</math>

|+ <math>\text{Table 43.}~~\text{Composite and Compiled Order Relations}</math>

|- style="background:#f0f0ff"

|- style="background:#f0f0ff"

Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.

Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.

But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so I want next to lay out some visual schemata that are designed to facilitate that.

But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so let us next lay out some visual schemata that are designed to facilitate that.

Figure 28-a shows the familiar picture of a boolean 3-cube, wherein the points of '''B'''³ are coordinated as bit strings of length three.  Looking at the functions ''f'' : '''B'''³ &rarr; '''B''' and the relations ''L'' &sube; '''B'''³ on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation ''L'' = <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> and which points are out of it.  Bowing to common convention, we may use the color "1" for points that are "in" a given relation and the color "0" for points that are "out" of that same relation.  However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

Figure&nbsp;28-a shows the familiar picture of a boolean 3-cube, where the points of <math>\mathbb{B}^3</math> are coordinated as bit strings of length three.  Looking at the functions <math>f : \mathbb{B}^3 \to \mathbb{B}</math> and the relations <math>L \subseteq \mathbb{B}^3</math> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation <math>L = [| f |]\!</math> and which points are out of it.  Bowing to common convention, we may use the color <math>1\!</math> for points that are ''in'' a given relation and the color <math>0\!</math> for points that are ''out'' of the same relation.  However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

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{| align="center" cellpadding="10" style="text-align:center; width:90%"