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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:

Finally, the set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a collection of three sets {X, Y, Z}, is called the "dyadic explosion" of {X, Y, Z}.  This object is denoted as "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2s" or simply as "X, Y, Z, choose 2", and is defined as follows:

Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).

| <math>\operatorname{Explo}(X, Y, Z ~|~ 2) ~=~ \operatorname{Pow}(X \times Y) \times \operatorname{Pow}(X \times Z) \times \operatorname{Pow}(Y \times Z).\!</math>

This domain is defined well enough for now to serve the immediate purposes of this section, but later it will be 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 suggestive identity,