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The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre-arranged order from a triple of three sets <math>(X, Y, Z),\!</math> is called the ''dyadic explosion'' of <math>X \times Y \times Z.\!</math>  This object is denoted <math>\operatorname{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\operatorname{choose}~ 2,\!</math> and defined as follows:

The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre-arranged order from a triple of three sets <math>(X, Y, Z),\!</math> is called the ''dyadic explosion'' of <math>X \times Y \times Z.\!</math>  This object is denoted <math>\operatorname{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\operatorname{choose}~ 2,\!</math> and defined as follows:

This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.

This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.

Just to provide a hint of what's at stake, consider the suggestive identity,

 

Just to provide a hint of what's at stake, consider the following suggestive identity:

2XY x 2XZ x 2YZ  = 2(XY + XZ + YZ),

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| <math>2^{XY} \times 2^{XZ} \times 2^{YZ} ~=~ 2^{(XY + XY + YZ)},\!</math>

and ask what sense would have to be found for the sums on the right in order to interpret this equation as a set theoretic isomorphism. Answering this question requires the concept of a "co product", roughly speaking, a "disjointed union" of sets.  By the time this discussion has detailed the forms of indexing necessary to maintain these constructions, it should have become patently obvious that the forms of analysis and synthesis that are called on to achieve the putative "reductions to" and "reconstructions from" dyadic relations in actual fact never really leave the realm of genuinely triadic relations, but merely reshuffle its contents in various convenient fashions.

What sense would have to be found for the sums on the right in order to interpret this equation as a set theoretic isomorphism? Answering this question requires the concept of a ''co-product'', roughly speaking, a &ldquo;disjointed union&rdquo; of sets.  By the time this discussion has detailed the forms of indexing necessary to maintain these constructions, it should have become patently obvious that the forms of analysis and synthesis that are called on to achieve the putative reductions to and reconstructions from dyadic relations in actual fact never really leave the realm of genuinely triadic relations, but merely reshuffle its contents in various convenient fashions.

==Scrap Area==

==Scrap Area==