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Thinking of relations in operational terms is facilitated by using a variant notation for tuples and sets of tuples, namely, the ordered pair (''x'', ''y'') is written ''x'':''y'', the ordered triple (''x'', ''y'', ''z'') is written ''x'':''y'':''z'', and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like ''a'':''b'' + ''b'':''c'' + ''c'':''d'' and so on.
Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on.
For example, translating the relations ''F'' ⊆ ''X'' × ''Y'' × ''Z'', ''G'' ⊆ ''X'' × ''Y'', ''H'' ⊆ ''Y'' × ''Z'' into this notation produces the following summary of the data:
For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data:
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}
|}
As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that ''G'' and ''H'' live in different spaces, is left implicit in the context of use.
As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use.
Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions:
Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions:
{| align="center" cellpadding="8" width="90%"
|
<math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math>
|}
Here is the big picture, with all of the pieces in place:
Here is the big picture, with all the pieces in place:
{| align="center" border="0" cellpadding="10"
{| align="center" border="0" cellpadding="10"
|}
|}
All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.
All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''F'' || = || 4:3:4 || + || 4:4:4 || + || 4:5:4
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}
|}
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''G'' ο ''H'' || = || (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4)
|
<math>\begin{matrix}
G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4)
\\[6pt]
& = & 4:4
\end{matrix}</math>
|}
|}
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''te''(''G'') || = || ''te''<sub>''XY''</sub><sup>''Z''</sup>(''G'')
|
<math>\begin{matrix}
\mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G)
\\[4pt]
& = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z)
\end{matrix}</math>
|}
|}
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''te''(''G'') || = || 4:3:1 || + || 4:4:1 || + || 4:5:1 || +
|
<math>\begin{matrix}
\mathrm{te}(G)
& = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\
& & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\
& & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\
& & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\
& & 4:3:7 & + & 4:4:7 & + & 4:5:7
\end{matrix}</math>
|}
|}
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''te''(''H'') || = || ''te''<sub>''YZ''</sub><sup>''X''</sup>(''H'')
|
<math>\begin{matrix}
\mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H)
\\[4pt]
& = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4)
\end{matrix}</math>
|}
|}
{| cellpadding=8 style="text-align:center"
{| align="center" cellpadding="8" width="90%"
| || ''te''(''H'') || = || 1:3:4 || + || 1:4:4 || + || 1:5:4 || +
|
<math>\begin{matrix}
\mathrm{te}(H)
& = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\
& & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\
& & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\
& & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\
& & 7:3:4 & + & 7:4:4 & + & 7:5:4
\end{matrix}</math>
|}
|}
{|
{| align="center" cellpadding="8" width="90%"
|
| align="center" | ''te''(''G'') ∩ ''te''(''H'')
<math>\begin{array}{ccl}
\mathrm{te}(G) \cap \mathrm{te}(H)
& = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4
\\[4pt]
G \circ H
& = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H))
\\[4pt]
& = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4)
\\[4pt]
& = & 4:4
\end{array}</math>
|}
|}
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* [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz]
* [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz]
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* [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web]
* [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web]
* [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis]
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* [http://planetmath.org/RelationComposition Relation Composition], [http://planetmath.org/ PlanetMath]
* [https://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [https://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia]
* [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia]
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