1. Functional Dependencies (Part 3) Presented by Nash Raghavan All page numbers are in reference to Database System Concepts (5 th Edition)

2. Agenda Quick Review What is a functional dependency Different normal forms Functional Dependency Theory Closure of a Set of Dependencies Decomposition using F.D. Multivalued dependencies

3. Quick Review What is a functional dependency? SSN → Name “Name is functionally dependent on SSN” Given a relation R, if   , then  must be the same whenever  is the same for all tuples in R First Normal Form Domains of all attributes in relation R are atomic Page 269 Boyce-Codd Normal Form For all functional dependencies   ,  is a superkey Page 272

4. Functional Dependency Theory Armstrong’s axioms (page 279) Reflexivity rule: If   , then  holds Example: A  A, BCD  BC Augmentation rule: If   , then    Example: A  B, therefore AC  BC Transitivity: If    and    then    Example: A  B and B  C then A  C

5. Functional Dependency Theory Additional rules (page 280) Union rule: If    and    then    Example: If A  B and A  C then A  BC Decomposition rule: If   , then    and    Example: A  BC, then A  B and A  C Pseudo-transitivity: If    and    then    Example: A  B and BD  C then AD  C

6. The Closure of a Set Why do we care about the axioms? Given a set of functional dependencies denoted by F { A  B, A  C, CG  H, CG  I, B  H } We can now determine logically implied functional dependencies such as A  H The set of all functional dependencies implied by F is denoted by F + and is called the closure of F See page 279 for more details

7. Decomposition The ability to compute F + enables us to convert a relation R into any normal form Example: BCNF Decomposition See pages 289 – 290 lending = ( branch_name, branch_city, assets, customer_name, loan_number, amount ) Candidate Key: { loan_number, customer_name } Functional Dependencies: branch_name  assets branch_city loan_number  amount branch_name

8. Decomposition The problem branch_name  assets branch_city Valid but branch_name is not a superkey, therefore it is not in BCNF! The solution – decomposition! lending = ( branch_name, branch_city, assets, customer_name, loan_number, amount ) Decomposes to multiple relations: branch = ( branch_name, branch_city, assets ) loan_info = ( branch_name, customer_name, loan_number, amount )

9. Decomposition branch relation is now in BCNF but loan_info is not because loan_number is not a superkey given the following dependency: loan_number  amount branch_name Solution is to iteratively decompose relations until all relations are in desired form. To complete solution, decompose loan_info to: loanb = ( loan_number, branch_name, amount ) borrower = ( customer_name, loan_number )

10. Decomposition The beginning: lending = ( branch_name, branch_city, assets, customer_name, loan_number, amount ) The end result: branch = ( branch_name, branch_city, assets ) loanb = ( loan_number, branch_name, amount ) borrower = ( customer_name, loan_number ) The Original Functional Dependencies: branch_name  assets branch_city loan_number  amount branch_name Everything is now in BCNF!

11. Multivalued Dependencies A multivalued dependency, denoted:    Means a tuple must exist for every value in  Example: class  books Class = { CS 157, CS 46 } Books = { Manual, Solution } CIDclassbooks 10CS 46Manual 10CS 46Solution 20CS 157Manual 20CS 157Solution Multivalued dependencies result in duplicate data and are considered “tuple-generating dependencies”. Formal definition – see page 295

12. The End This information will become relevant … eventually. Makes more sense when we study: Database design/creation Normalization BCNF, 1NF, 2NF, 3NF, 4NF Decomposition Denormalization