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In the special case of a dyadic relation <math>L \subseteq X_1 \times X_2 = X \times Y,\!</math> including the case of a partial function <math>p : X \rightharpoonup Y\!</math> or a total function <math>f : X \to Y,\!</math> we have the following conventions.

In the special case of a dyadic relation R c X1xX2 = SxT, including the case of a partial function p : S ~> T or a total function f : S  > T, I will stick to the following conventions:

1. The arbitrarily designated domains X1 = S and X2 = T that form the widest sets admitted to the dyadic relation are referred to as the "domain" or "source" and the "codomain" or "target", respectively, of the relation in question.

# The arbitrarily designated domains <math>X_1 = X\!</math> and <math>X_2 = Y\!</math> that form the widest sets admitted to the dyadic relation are referred to as the ''domain'' or ''source'' and the ''codomain'' or ''target'', respectively, of the relation in question.

 

# The terms ''quota'' and ''range'' are reserved for those uniquely defined sets whose elements actually appear as the first and second members, respectively, of the ordered pairs in that relation.  Thus, for a dyadic relation <math>L \subseteq X \times Y,\!</math> we identify <math>\operatorname{Quo} (L) = \operatorname{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\operatorname{Ran} (L) = \operatorname{Quo}_2 (L) \subseteq Y\!</math> with the usual ''range'' of <math>L.\!</math>

2. The terms "quota" and "range" are reserved for those uniquely defined sets whose elements actually appear as the 1st and 2nd members, respectively, of the ordered pairs in that relation.  Thus, for a dyadic relation R c SxT, I let Quo (R) = Quo1 (R) c S be identified with what is usually called the "domain of definition" of R, and I let Ran (R) = Quo2 (R) c T be identified with the usual range of R.

A "partial equivalence relation" (PER) on a set X is a relation R c XxX that is an equivalence relation on its domain of definition Quo (R) c X.  In this situation, [x]R is empty for each x in X that is not in Quo (R).  Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive.  Like the "self identical elements" of old that epitomized the very definition of self consistent existence in classical logic, the property of being a self related or self equivalent element in the purview of a PER on X singles out the members of Quo (R) as those for which a properly meaningful existence can be contemplated.

A "partial equivalence relation" (PER) on a set X is a relation R c XxX that is an equivalence relation on its domain of definition Quo (R) c X.  In this situation, [x]R is empty for each x in X that is not in Quo (R).  Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive.  Like the "self identical elements" of old that epitomized the very definition of self consistent existence in classical logic, the property of being a self related or self equivalent element in the purview of a PER on X singles out the members of Quo (R) as those for which a properly meaningful existence can be contemplated.