Editing Functional dependency (section)

== Covers and equivalence ==

=== Covers ===
'''Definition''': <math>F</math> covers <math>G</math> if every FD in <math>G</math> can be inferred from <math>F</math>. <math>F</math> covers <math>G</math> if <math>G</math><sup>+</sup> &sube; <math>F</math><sup>+</sup> <br/>
Every set of functional dependencies has a [[canonical cover]].

=== Equivalence of two sets of FDs ===
Two sets of FDs <math>F</math> and <math>G</math> over schema <math>R</math> are equivalent, written <math>F</math> &equiv; <math>G</math>, if <math>F</math><sup>+</sup> = <math>G</math><sup>+</sup>. If <math>F</math> &equiv; <math>G</math>, then <math>F</math> is a cover for <math>G</math> and vice versa. In other words, equivalent sets of functional dependencies are called ''covers'' of each other.

=== Non-redundant covers ===
A set <math>F</math> of FDs is nonredundant if there is no proper subset 
<math>F'</math> of <math>F</math> with <math>F'</math> &equiv; <math>F</math>. If such an <math>F'</math> exists, <math>F</math> is redundant. <math>F</math> is a nonredundant cover for <math>G</math> if <math>F</math> is a cover for <math>G</math> and <math>F</math> is nonredundant.
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An alternative characterization of nonredundancy is that <math>F</math> is nonredundant if there is no FD ''X'' → ''Y'' in <math>F</math> such that <math>F </math> - {''X'' → ''Y''} <math>\models</math> ''X'' → ''Y''. Call an FD ''X'' → ''Y'' in <math>F</math> redundant in <math>F</math> if <math>F </math> - {''X'' → ''Y''} <math>\models</math> ''X'' → ''Y''.