Finitary relation

Consider the relation S involving three roles that people might play, that is expressed in a statement of the form "X thinks that Y likes Z ". For simplicity, we notate a set of possible scenarios in a table as follows:

Here, each row of the table makes an assertion of the form "X thinks that Y likes Z ". For instance, the first row asserts that "Alice thinks that Bob likes Denise". The table represents a relation S over the set P, the set of all people under discussion. That is:

P = {Alice, Bob, Charles, Denise}.

The data of the table are equivalent to the following set of ordered triples:

S = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

The notation S(Alice, Bob, Denise) has the same effect as the first row of the table. The relation S is a ternary relation, since there are three items involved in each row. The relation is itself a set-theoretic object defined as a subset of the Cartesian product P×P×P, where P is the set of individuals as discussed above. Thus mathematically, a relation is simply a collection of "ordered list".

The above table for relation S is also a simple example of a relational database, a field with theory rooted in relational algebra and applications in data management.[4] Computer scientists, logicians, and mathematicians, however, tend to have different conceptions what a general relation is, and what it is consisted of. For example, databases are designed to deal with empirical data, which is by definition finite, whereas in mathematics, relations with infinite dimensions (i.e., infinitary relation) are also considered.

The non-negative integer ${\displaystyle k}$ giving the number of "places" in the relation, 3 for the above example, is called the relation's arity, rank, adicity, or dimension. A relation with ${\displaystyle k}$ places is variously called a ${\displaystyle k}$-ary, a ${\displaystyle k}$-adic, or a ${\displaystyle k}$-dimensional relation. Relations with a finite number of places are called finite-place or finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to include infinitary relations between infinitudes of numbers,[5] as in the case of infinite sequences.

Since there is only one 0-tuple, the empty tuple ( ), there are only two zero-place relations: the one that always holds, and the one that never holds. These are sometimes useful for constructing the base case of an induction argument. One-place relations are called unary relations. For instance, any set (such as the collection of Nobel laureates) can be viewed as a collection of members having some property (such as that of having been awarded the Nobel prize). Two-place relations are called binary relations or, in some cases, dyadic relations.[6] Binary relations are the most commonly studied form of finitary relations,[1] whose examples include, among others:

• Equality and inequality, denoted by signs such as '${\displaystyle =}$' and '${\displaystyle <}$' in statements such as '${\displaystyle 5<12}$';
• Divisibility, denoted by the sign '${\displaystyle \mid }$' in statements such as '${\displaystyle 13\mid 143}$';
• Set membership, denoted by the sign '${\displaystyle \in }$' in statements such as '${\displaystyle 1\in \mathbb {N} }$'.

A ${\displaystyle k}$-ary relation is a generalization of a binary relation.

When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation.

— Augustus De Morgan[7]

The simpler of the two definitions of ${\displaystyle k}$-place relations encountered in mathematics is:

Definition 1. A relation L over the sets X₁, …, X_k is a subset of their Cartesian product, written L ⊆ X₁ × … × X_k.

Relations are classified according to the number of sets in the defining Cartesian product, in other words, according to the number of terms following L. Hence:

• Lu denotes a unary relation or property;
• Luv or uLv denotes a binary relation (an homogeneous relation when X₁ = X₂ and a heterogeneous relation when X₁ ≠ X₂);
• Luvw denotes a ternary relation;
• Luvwx denotes a quaternary relation.

Relations with more than four terms are usually referred to as k-ary or n-ary, for example, a "5-ary relation". A k-ary relation is simply a set of k-tuples.[1]

The second definition makes use of an idiom that is common in mathematics, stipulating that "such and such is an n-tuple" in order to ensure that such and such a mathematical object is determined by the specification of n-component mathematical objects. In the case of a relation L over k sets, there are k + 1 things to specify, namely, the k sets plus a subset of their Cartesian product. In the idiom, this is expressed by saying that L is a (k + 1)-tuple.

Definition 2. A relation L over the sets X₁, …, X_k is a (k + 1)-tuple L = (X₁, …, X_k, G(L)), where G(L) is a subset of the Cartesian product X₁ × … × X_k. G(L) is called the graph of L.

Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element a = (a₁, …, a_k) or the variable element x = (x₁, …, x_k).

A statement of the form "a is in the relation L " or "a satisfies L " is taken to mean that a is in L under the first definition, and that a is in G(L) under the second definition.

The following considerations apply under either definition:

• The sets X_j for j = 1 to k are called the domains of the relation. Under the first definition, the relation does not uniquely determine a given sequence of domains.
• If all of the domains X_j are the same set X, then it is simpler to refer to L as a k-ary relation over X (i.e., a homogeneous relation).
• If any of the domains X_j is empty, then the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation L = ${\displaystyle \varnothing }$. Hence it is commonly stipulated that all of the domains be nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and if it ever becomes necessary to distinguish between the two definitions, an entity satisfying the second definition may be called an embedded or included relation.

If L is a relation over the domains X₁, …, X_k, it is conventional to consider a sequence of terms called variables, x₁, …, x_k, that are said to range over the respective domains.

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 the relation L, written ƒ_L or χ(L), is the Boolean-valued function ƒ_L : X₁ × … × X_k → B, defined in such a way that ƒ_L(${\displaystyle \mathbf {x} }$) = 1 precisely when the k-tuple ${\displaystyle \mathbf {x} }$ is in the relation L. Such a function can also be called an indicator function, particularly in probability and statistics, to avoid confusion with the notion of a characteristic function in probability theory.

In applied mathematics, computer science and statistics, it is common to refer to a Boolean-valued function (such as ƒ_L) as a k-place predicate. From the more abstract viewpoint of formal logic and model theory, the relation L constitutes a logical model or a relational structure, that serves as one of many possible interpretations of some k-place predicate symbol.

Because relations arise in many scientific disciplines, as well as in many branches of mathematics and logic, there is considerable variation in terminology. Aside from the set-theoretic extension of a relational concept or term, the term "relation" can also be used to refer to the corresponding logical entity, either the logical comprehension, which is the totality of intensions or abstract properties shared by all elements in the relation, or else the symbols denoting these elements and intensions. Further, some writers of the latter persuasion introduce terms with more concrete connotations (such as "relational structure" for the set-theoretic extension of a given relational concept).

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990). Charles Sanders Peirce restated and extended De Morgan's results.

In the 19th century, Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind and others advanced the theory of relations. Many of their ideas, especially on relations called orders, were summarized in Principles of Mathematics (1903), where Bertrand Russell and Alfred Whitehead, made free use of these results.

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