A relation of degree zero, 0-ary relation, or

nullary Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...

relation is a relation with zero attributes. There are exactly two relations of degree zero. One has cardinality zero; that is, contains no

tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

s at all. The other has cardinality 1 contains the unique 0-tuple.^:56 The zero-degree relations represent true and false in

relational algebra In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory was introduced by Edgar F. Codd. The main application of relational algebr ...

.^:57 Under the

closed-world assumption The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. T ...

, an ''n''-ary relation is interpreted as the extension of some ''n''-adic

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...

: all and only those ''n''-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions that hold true, appear in the relation. A zero-degree relation is therefore interpreted as the extension of the 0-adic predicate . The zero-degree relation with cardinality zero therefore represents false because it contains no tuples that yield a true proposition, and the zero-degree relation with cardinality 1 represents true because it contains the unique 0-tuple that yields a true proposition. The zero-degree relations are also significant as identities for certain operators in the relational algebra. The zero-degree relation of cardinality 1 is the identity with respect to

join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...

(⋈); that is, when it is joined with any other relation , the result is . Defining an identity with respect to join makes it possible to extend the

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...

join operator into a ''n''-ary join operator.^:89 Since the relational Cartesian product is a special case of join, the zero-degree relation of cardinality 1 is also the identity with respect to the Cartesian product.^:89 A projection of a relation over no attributes yields one of the relations of degree zero. If the projected relation has cardinality 0, the projection will have cardinality 0; if the projected relation has positive cardinality, the result will have cardinality 1.

Hugh Darwen Hugh Darwen is a computer scientist who was an employee of IBM United Kingdom from 1967. to 2004, and has been involved in the development of the relational model. Work From 1978 to 1982 he was a chief architect on Business System 12, a dat ...

refers to the zero-degree relation with cardinality 0 as TABLE_DUM and the relation with cardinality 1 as TABLE_DEE, alluding to the characters

Tweedledum and Tweedledee Tweedledum and Tweedledee are characters in an English nursery rhyme and in Lewis Carroll's 1871 book ''Through the Looking-Glass, and What Alice Found There''. Their names may have originally come from an epigram written by poet John Byrom. The ...

References

{{Reflist Relational algebra

See Also

References