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# The geometric language of sets

# The geometric language of sets

Thus, one way to explain the relationships that hold among these concepts is to describe the ''translations'' that are induced among the allied families of notation.

Thus, one way to explain the relationships that hold among these concepts is to describe the ''translations'' that are induced among their allied families of notation.

===Syntactic Transformation Rules===

=====1.3.12.1.  Syntactic Transformation Rules=====

A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them.  A rudimentary example of a STR is readily mined from the raw materials that are already available in this area of discussion.  To begin, let the definition of an indicator function be recorded in the following form:

A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them.  A rudimentary example of a STR is readily mined from the raw materials that are already available in this area of discussion.  To begin, let the definition of an indicator function be recorded in the following form:

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===Derived Equivalence Relations===

=====1.3.12.2.  Derived Equivalence Relations=====

One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations.  With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies.  Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.

One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations.  With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies.  Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.

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===Digression on Derived Relations===

=====1.3.12.3.  Digression on Derived Relations=====

A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> with other types of operations on triadic relations.  The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.

A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> with other types of operations on triadic relations.  The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.