Discuss the purpose of Armstrong’s axioms.

Question

What will be an ideal response?

Answer 1

The set of all functional dependencies that are implied by a given set of functional
dependencies X is called the closure of X, written X ? . We clearly need a set of rules to help
compute X ? from X. A set of inference rules, called Armstrong’s axioms, specifies how new
functional dependencies can be inferred from given ones (Armstrong, 1974). For our
discussion, let A, B, and C be subsets of the attributes of the relation R. Armstrong’s axioms
are as follows:
(1) Reflexivity: If B is a subset of A, then A ? B
(2) Augmentation: If A ? B, then A,C ? B,C
(3) Transitivity: If A ? B and B ? C, then A ? C
Note that each of these three rules can be directly proved from the definition of functional
dependency. The rules are complete in that given a set X of functional dependencies, all
functional dependencies implied by X can be derived from X using these rules. The rules are
also sound in that no additional functional dependencies can be derived that are not implied
by X. In other words, the rules can be used to derive the closure of X ? .
Several further rules can be derived from the three given above that simplify the practical task
of computing X
To begin to identify the set of functional dependencies F for a relation, typically we first
identify the dependencies that are determined from the semantics of the attributes of the
relation. Then, we apply Armstrong’s axioms (Rules 1 to 3) to infer additional functional
dependencies that are also true for that relation. A systematic way to determine these additional
functional dependencies is to first determine each set of attributes A that appears on the left-
hand side of some functional dependencies and then to determine the set of all attributes that
are dependent on A. Thus, for each set of attributes A we can determine the set A ? of attributes