Changes

The power set notation can be used to provide an alternative description of relations.  In the case where <math>S\!</math> is a cartesian product, say <math>S = X_1 \times \ldots \times X_n,\!</math> then each <math>n\!</math>-place relation <math>L\!</math> described as a subset of <math>S,\!</math> say <math>L \subseteq X_1 \times \ldots \times X_n,\!</math> is equally well described as an element of <math>\operatorname{Pow}(S),\!</math> in other words, as <math>L \in \operatorname{Pow}(X_1 \times \ldots \times X_n).\!</math>

The power set notation can be used to provide an alternative description of relations.  In the case where <math>S\!</math> is a cartesian product, say <math>S = X_1 \times \ldots \times X_n,\!</math> then each <math>n\!</math>-place relation <math>L\!</math> described as a subset of <math>S,\!</math> say <math>L \subseteq X_1 \times \ldots \times X_n,\!</math> is equally well described as an element of <math>\operatorname{Pow}(S),\!</math> in other words, as <math>L \in \operatorname{Pow}(X_1 \times \ldots \times X_n).\!</math>

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 by <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 (X, Y, Z), is called the ''dyadic explosion'' of XxYxZ.  This object is denoted by "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by twos", or more simply as "X, Y, Z, choose 2", and defined as follows:</p>

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| <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>

| <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>

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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.

By means of these constructions the operation that forms Proj (R) for each triadic relation R c XxYxZ can be expressed as a function:

By means of these constructions the operation that forms Proj (R) for each triadic relation R c XxYxZ can be expressed as a function: