Editing Relational model (section)

== Logical interpretation ==

The relational model is a [[formal system]]. A relation's attributes define a set of [[logic]]al [[propositions]]. Each proposition can be expressed as a tuple. The body of a relation is a subset of these tuples, representing which propositions are true. Constraints represent additional propositions which must also be true. [[Relational algebra]] is a set of logical rules that can [[Validity (logic)|validly]] [[inference|infer]] conclusions from these propositions.<ref name="professionals"/>{{rp|95–101}}

The definition of a ''tuple'' allows for a unique empty tuple with no values, corresponding to the [[empty set]] of attributes. If a relation has a degree of 0 (i.e. its heading contains no attributes), it may have either a cardinality of 0 (a body containing no tuples) or a cardinality of 1 (a body containing the single empty tuple). These relations represent [[Boolean domain|Boolean]] [[truth values]]. The relation with degree 0 and cardinality 0 is ''False'', while the relation with degree 0 and cardinality 1 is ''True''.<ref name="professionals"/>{{rp|221–223}}

=== Example ===

If a relation of Employees contains the attributes ''{Name, ID}'', then the tuple ''{Alice, 1}'' represents the proposition: "There exists an employee named ''Alice'' with ID ''1''". This proposition may be true or false. If this tuple exists in the relation's body, the proposition is true (there is such an employee). If this tuple is not in the relation's body, the proposition is false (there is no such employee).<ref name="professionals"/>{{rp|96–97}}

Furthermore, if ''{ID}'' is a key, then a relation containing the tuples ''{Alice, 1}'' and ''{Bob, 1}'' would represent the following [[contradiction]]:
# There exists an employee with the name ''Alice'' and the ID ''1''.
# There exists an employee with the name ''Bob'' and the ID ''1''.
# There do not exist multiple employees with the same ID.

Under the [[principle of explosion]], this contradiction would allow the system to prove that any arbitrary proposition is true. The database must enforce the key constraint to prevent this.<ref name="professionals"/>{{rp|104}}