Editing Functional dependency (section)

== Closure ==

=== Closure of functional dependency ===
The closure of a set of values is the set of attributes that can be determined using its functional dependencies for a given relationship. One uses [[Armstrong's axioms]] to provide a proof - i.e. reflexivity, augmentation, transitivity.

Given <math>R</math> and <math>F</math> a set of FDs that holds in <math>R</math>:
The closure of <math>F</math> in <math>R</math> (denoted <math>F</math><sup>+</sup>) is the set of all FDs that are logically implied by <math>F</math>.<ref>{{Cite journal|last=Saiedian|first=H.|date=1996-02-01|title=An Efficient Algorithm to Compute the Candidate Keys of a Relational Database Schema|url=https://academic.oup.com/comjnl/article-lookup/doi/10.1093/comjnl/39.2.124|journal=The Computer Journal|language=en|volume=39|issue=2|pages=124–132|doi=10.1093/comjnl/39.2.124|issn=0010-4620}}</ref>

=== Closure of a set of attributes ===
Closure of a set of attributes X with respect to <math>F</math> is the set X<sup>+</sup> of all attributes that are functionally determined by X using <math>F</math><sup>+</sup>.

==== Example ====
Imagine the following list of FDs. We are going to calculate a closure for A (written as A<sup>+</sup>) from this relationship.

# ''A'' → ''B''
# ''B'' → ''C''
# ''AB'' → ''D''

The closure would be as follows:
{{ordered list | list-style-type = lower-alpha
| A → A (by Armstrong's reflexivity)
| A → AB (by 1. and (a))
| A → ABD (by (b), 3, and Armstrong's transitivity)
| A → ABCD (by (c), and 2)
}}
Therefore, A<sup>+</sup>= ABCD. Because A<sup>+</sup> includes every attribute in the relationship, it is a [[superkey]].