In mathematics, a nullary relation, 0-ary relation, or relation of degree zero 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.<sup>:56</sup>
The zero-degree relations represent true and false in relational algebra.<sup>:57</sup> 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 (âÂÂ); 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.<sup>:89</sup>
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.<sup>:89</sup>
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.