Relation Theory • 3

Relation TheoryDefinition

It is convenient to begin with the definition of a k-place relation, where k is a positive integer.

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

Several items of terminology are useful in discussing relations.

  • The sets X_1, \ldots, X_k are called the domains of the relation L \subseteq X_1 \times \ldots \times X_k, with {X_j} being the j^\text{th} domain.
  • If all the {X_j} are the same set X then L \subseteq X_1 \times \ldots \times X_k is more simply described as a
    k-place relation over X.
  • The set L is called the graph of the relation L \subseteq X_1 \times \ldots \times X_k, on analogy with the graph of a function.
  • If the sequence of sets X_1, \ldots, X_k is constant throughout a given discussion or is otherwise determinate in context then the relation L \subseteq X_1 \times \ldots \times X_k is determined by its graph L, making it acceptable to denote the relation by referring to its graph.
  • Other synonyms for the adjective k-place are k-adic and k-ary, all of which leads to the integer k being called the dimension, adicity, or arity of the relation L.


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7 Responses to Relation Theory • 3

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