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Revision as of 14:42, 15 April 2013
, 14:42, 15 April 2013→6.35. Reducibility of Sign Relations
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 <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:
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