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

nullary In logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the ...

relation is a

relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

with zero attributes. There are exactly two relations of degree zero. One has

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

zero; that is, contains no

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

s at all. The other has cardinality 1 and contains only 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 for modeling data and defining queries on it with well founded semantics (computer science), semantics. The theory was introduced by Edgar F. Codd. The main applica ...

.^:57 Under the closed-world assumption, an ''n''-ary relation is interpreted as the extension of some ''n''-adic predicate: 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 topo ...

(⋈); 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 join operator into a ''n''-ary join operator.^:89 Since the relational

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

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 Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...

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. T ...

References

{{Reflist Relational algebra

See also

References