In the following example, Fc is the canonical cover of F.
Given the following, we can find the canonical cover: R = (A, B, C, G, H, I), F = {A→BC, B→C, A→B, AB→C}
{A→BC, B→C, A→B, AB→C}
{A → BC, B →C, AB → C}
{A → BC, B → C}
{A → B, B →C}
Fc = {A → B, B →C}
Extraneous attributes
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An attribute is extraneous in a functional dependency if its removal from that functional dependency does not alter the closure of any attributes.[2]
Extraneous determinant attributes
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Given a set of functional dependencies and a functional dependency in , the attribute is extraneous in if and any of the functional dependencies in can be implied by using Armstrong's Axioms.[2]
Using an alternate method, given the set of functional dependencies , and a functional dependency X → A in , attribute Y is extraneous in X if , and .[3]
For example:
If F = {A → C, AB → C}, B is extraneous in AB → C because A → C can be inferred even after deleting B. This is true because if A functionally determines C, then AB also functionally determines C.
If F = {A → D, D → C, AB → C}, B is extraneous in AB → C because {A → D, D → C, AB → C} logically implies A → C.
Extraneous dependent attributes
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Given a set of functional dependencies and a functional dependency in , the attribute is extraneous in if and any of the functional dependencies in can be implied by using Armstrong's axioms.[3]
A dependent attribute of a functional dependency is extraneous if we can remove it without changing the closure of the set of determinant attributes in that functional dependency.[2]
For example:
If F = {A → C, AB → CD}, C is extraneous in AB → CD because AB → C can be inferred even after deleting C.
If F = {A → BC, B → C}, C is extraneous in A → BC because A → C can be inferred even after deleting C.
References
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^Silberschatz, Abraham (2011). Database system concepts(PDF) (Sixth ed.). New York: McGraw-Hill. ISBN 978-0073523323. Archived from the original (PDF) on 2020-11-08.
^ abcElmasri, Ramez (2016). Fundamentals of database systems. Sham Navathe (Seventh ed.). Hoboken, NJ: Pearson. ISBN 978-0-13-397077-7. OCLC 913842106.
^ abK, Saravanakumar; asamy. "How to find extraneous attribute? An example". Retrieved 2023-03-14.