# A a a 1 2 n b a a a 12 nin english though a bit vague

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A , A , … A 1 2 n B A , A , … A 1 2 n In English (though a bit vague): Whenever a set of attributes of R is determining another attribute, should determine all the attributes of R .
Example Name SSN Phone Number Fred 123-321-99 (201) 555-1234 Fred 123-321-99 (206) 572-4312 Joe 909-438-44 (908) 464-0028 Joe 909-438-44 (212) 555-4000 What are the dependencies? SSN Name What are the keys? Is it in BCNF?
Decompose it into BCNF SSN Name 123-321-99 Fred 909-438-44 Joe SSN Phone Number 123-321-99 (201) 555-1234 123-321-99 (206) 572-4312 909-438-44 (908) 464-0028 909-438-44 (212) 555-4000 SSN Name
What About This? Name Price Category Gizmo \$19.99 gadgets OneClick \$24.99 camera Name Price, Category
BCNF Decomposition Find a dependency that violates the BCNF condition: A , A , … A 1 2 n B , B , … B 1 2 m A’s Others B’s R1 R2 Heuristics: choose B , B , … B “as large as possible” 1 2 m Decompose: Find a 2-attribute relation that is not in BCNF. Continue until there are no BCNF violations left.
Example Decomposition Name SSN Age EyeColor PhoneNumber Functional dependencies: SSN Name, Age, Eye Color What if we also had an attribute Draft-worthy, and the FD: Age Draft-worthy Person: BNCF: Person1( SSN, Name, Age, EyeColor ), Person2( SSN, PhoneNumber )
Other Example R( A,B,C,D ) A B, B C Key: Violations of BCNF: Pick : split into R1( ) R2( )
Correct Decompositions A decomposition is lossless if we can recover: R(A,B,C) R1(A,B) R2(A,C) R’(A,B,C) = R(A,B,C) R’ is in general larger than R. Must ensure R’ = R

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• Fall '19
• Database normalization

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