# Armstrong Axioms

**DOI:**https://doi.org/10.1007/978-1-4899-7993-3_1554-2

## Definition

The term *Armstrong axioms* refers to the sound and complete set of inference rules or axioms, introduced by William W. Armstrong [2], that is used to test logical implication of functional dependencies.

Given a relation schema *R*[*U*] and a set of functional dependencies Σ over attributes in *U*, a functional dependency *f* is logically implied by Σ, denoted by Σ⊧*f*, if for every instance *I* of *R* satisfying all functional dependencies in Σ, *I* satisfies *f*. The set of all functional dependencies implied by Σ is called the *closure* of Σ, denoted by Σ^{+}.

## Key Points

*Reflexivity*: If*Y*⊆*X*, then*X*→*Y*.*Augmentation*: If*X*→*Y*, then*XZ*→*YZ*.*Transitivity:*If*X*→*Y*and*Y*→*Z*, then*X*→*Z*.

Note that in the above rules *XZ* refers to the union of two attribute sets *X* and *Z*. Armstrong axioms are *sound* and *complete*: a functional dependency *f* is derivable from a set of functional dependencies Σ by applying the axioms if and only if Σ⊧*f* (refer to [1]...

## Recommended Reading

- 1.Abiteboul S, Hull R, Vianu V. Foundations of databases. Reading: Addison-Wesley; 1995.Google Scholar
- 2.Armstrong W. Dependency structures of data base relationships. In: Proceedings of the IFIP Congress; 1974.Google Scholar