relation of degree zero

{{Short description|A relation with zero attributes}}

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A relation of degree zero, 0-ary relation, or nullary relation is a relation with zero attributes. There are exactly two relations of degree zero. One has cardinality zero; that is, contains no tuples at all. The other has cardinality 1 and contains only the unique 0-tuple.{{cite book |last1=Date |first1=Christopher J. |title=Database In Depth |date=May 2005 |publisher=O'Reilly |location=Sebastopol, California |isbn=978-0-596-10012-4 |edition=1}}^:56

The zero-degree relations represent true and false in relational algebra.^: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 {{math|P() → true}}. 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 (⋈); that is, when it is joined with any other relation {{math|R}}, the result is {{math|R}}. 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 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 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.{{Cite encyclopedia|first=Hugh|last=Darwen|editor1-last=Date|editor1-first=C. J.|editor2-last=Darwen|editor2-first=Hugh|title=The Nullologist in Relationland|encyclopedia=Relational Database Writings 1989–1991|year=1992|publisher=Addison-Wesley|location=Reading, Massachusetts}}