Relation System

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A Relation System is a Formal Logical System that is defined by Relation Functions and Set Operation.



  • (Wikiversity, 2019) ⇒ 2019-09-25
    • QUOTE: It is convenient to begin with the definition of a [math]k\![/math]-place relation, where [math]k\![/math] is a positive integer.

      Definition. A [math]k\![/math]-place relation [math]L \subseteq X_1 \times \ldots \times X_k\![/math] over the nonempty sets [math]X_1, \ldots, X_k\![/math] is a [math](k+1)\![/math]-tuple [math](X_1, \ldots, X_k, L)\![/math] where [math]L\![/math] is a subset of the cartesian product [math]X_1 \times \ldots \times X_k.\![/math]. (...)

      Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets [math]X_1, \ldots, X_k\![/math] are called the domains of the relation [math]L \subseteq X_1 \times \ldots \times X_k,\![/math] with [math]{X_j}\![/math] being the [math]j^\text{th}\![/math] domain. If all of the [math]{X_j}\![/math] are the same set [math]X\![/math] then [math]L \subseteq X_1 \times \ldots \times X_k\![/math] is more simply described as a [math]k\![/math]-place relation over [math]X.\![/math] The set [math]L\![/math] is called the graph of the relation [math]L \subseteq X_1 \times \ldots \times X_k,\![/math] on analogy with the graph of a function. If the sequence of sets [math]X_1, \ldots, X_k\![/math] is constant throughout a given discussion or is otherwise determinate in context, then the relation [math]L \subseteq X_1 \times \ldots \times X_k\![/math] is determined by its graph [math]L,\![/math] making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective [math]k\![/math]-place are [math]k\![/math]-adic and [math]k\![/math]-ary, all of which leads to the integer [math]k\![/math] being called the dimension, adicity, or arity of the relation [math]L.\![/math] (...)

      Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject.

      One dimension of variation is reflected in the names that are given to [math]k\![/math]-place relations, for [math]k = 0, 1, 2, 3, \ldots,\![/math] with some writers using the Greek forms, medadic, monadic, dyadic, triadic, [math]k\![/math]-adic, and other writers using the Latin forms, nullary, unary, binary, ternary, [math]k\![/math]-ary.

      The number of relational domains may be referred to as the adicity, arity, or dimension of the relation. Accordingly, one finds a relation on a finite number of domains described as a polyadic relation or a finitary relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to [math]k,\![/math] then the relation may be described as a [math]k\![/math]-adic relation, a [math]k\![/math]-ary relation, or a [math]k\![/math]-dimensional relation, respectively.

      A more conceptual than nominal variation depends on whether one uses terms like predicate, relation, and even term to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated.