Changes

{| align="center" cellspacing="8" width="90%"

{| align="center" cellspacing="8" width="90%"

| <math>\operatorname{Proj}^{(2)}(L) ~=~ (\operatorname{proj}_{12}(L), \operatorname{proj}_{13}(L), \operatorname{proj}_{23}(L)).</math>

| <math>\operatorname{Proj}^{(2)} L ~=~ (\operatorname{proj}_{12} L, ~ \operatorname{proj}_{13} L, ~ \operatorname{proj}_{23} L).\!</math>

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If <math>L\!</math> is visualized as a solid body in the 3-dimensional space <math>X \times Y \times Z,\!</math> then <math>\operatorname{Proj}^{(2)}(L)\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY,\!</math> <math>XZ,\!</math> and <math>YZ\!</math> planes, respectively.

If <math>L\!</math> is visualized as a solid body in the 3-dimensional space <math>X \times Y \times Z,\!</math> then <math>\operatorname{Proj}^{(2)} L\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY, ~ XZ, ~ YZ\!</math> planes, respectively.

A couple of set-theoretic constructions are worth introducing at this point, in particular for describing the source and target domains of the projection operator <math>\operatorname{Proj}^{(2)}.\!</math>

Two more set-theoretic constructions are worth introducing at this point, in particular for describing the source and target domains of the projection operator <math>\operatorname{Proj}^{(2)}.\!</math>

The set of subsets of a set <math>S\!</math> is called the ''power set'' of <math>S.\!</math> This object is denoted by either of the forms <math>\operatorname{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:

1. The set of subsets of a set S is called the "power set" of S.  This object is denoted by either of the forms "Pow (S)" or "2S" and defined as follows:

Pow (S) = 2S  = {T : T c S}.

| <math>\operatorname{Pow}(S) ~=~ 2^S ~=~ \{ T : T \subseteq S \}.\!</math>

The power set notation can be used to provide an alternative description of relations.  In the case where S is a cartesian product, say S = X1x...xXn, then each n place relation described as a subset of S, say as R c X1x...xXn, is equally well described as an element of Pow (S), in other words, as R C Pow (X1x...xXn).

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>

2. 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 2s", or more simply as "X, Y, Z, choose 2", 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>

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 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.