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Finitary relation
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Finitary relation
In mathematics, a finitary relation over a sequence of sets X1, ..., Xn is a subset of the Cartesian product X1 × ... × Xn; that is, it is a set of n-tuples (x1, ..., xn), each being a sequence of elements xi in the corresponding Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true.
The non-negative integer n that gives the number of "places" in the relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences.
When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation.
Since the definition is predicated on the underlying sets X1, ..., Xn, R may be more formally defined as the (n + 1)-tuple (X1, ..., Xn, G), where G, called the graph of R, is a subset of the Cartesian product X1 × ... × Xn.
As is often done in mathematics, the same symbol is used to refer to the mathematical object and an underlying set, so the statement (x1, ..., xn) ∈ R is often used to mean (x1, ..., xn) ∈ G is read "x1, ..., xn are R-related" and are denoted using prefix notation by Rx1⋯xn and using postfix notation by x1⋯xnR. In the case where R is a binary relation, those statements are also denoted using infix notation by x1Rx2.
The following considerations apply:
Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values, typically 0 = false and 1 = true. The characteristic function of R, denoted by χR, is the Boolean-valued function χR: X1 × ... × Xn → B, defined by χR((x1, ..., xn)) = 1 if Rx1⋯xn and χR((x1, ..., xn)) = 0 otherwise.
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Finitary relation
In mathematics, a finitary relation over a sequence of sets X1, ..., Xn is a subset of the Cartesian product X1 × ... × Xn; that is, it is a set of n-tuples (x1, ..., xn), each being a sequence of elements xi in the corresponding Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true.
The non-negative integer n that gives the number of "places" in the relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences.
When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation.
Since the definition is predicated on the underlying sets X1, ..., Xn, R may be more formally defined as the (n + 1)-tuple (X1, ..., Xn, G), where G, called the graph of R, is a subset of the Cartesian product X1 × ... × Xn.
As is often done in mathematics, the same symbol is used to refer to the mathematical object and an underlying set, so the statement (x1, ..., xn) ∈ R is often used to mean (x1, ..., xn) ∈ G is read "x1, ..., xn are R-related" and are denoted using prefix notation by Rx1⋯xn and using postfix notation by x1⋯xnR. In the case where R is a binary relation, those statements are also denoted using infix notation by x1Rx2.
The following considerations apply:
Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values, typically 0 = false and 1 = true. The characteristic function of R, denoted by χR, is the Boolean-valued function χR: X1 × ... × Xn → B, defined by χR((x1, ..., xn)) = 1 if Rx1⋯xn and χR((x1, ..., xn)) = 0 otherwise.