Relation of degree zero

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 contains the unique 0-tuple. :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 $P → true$. 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 $R$, the result is $R$. 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.