1. Chapter 5.1 and 5.2 Brian Cobarrubia Database Management Systems II January 31, 2008

2. Agenda 5.1 – Functional Dependencies Between Attributes  Superkeys and Keys  Inferring Functional Dependencies  Determining Keys from Functional Dependencies 5.2 – Normal Forms  1 st Normal Form  2 nd Normal Form  3 rd Normal Form  Boyce Codd Normal Form

3. Functional Dependencies between Attributes Functionally Dependent  An attribute B is functionally dependent on an attribute A in a relational table if the value of A uniquely determines the value of B  We can also say that A determines B A  B  Formal Definition of Functional Dependencies Let R be a relation schema where   R and   R The functional dependency    Holds on R if and only if for any legal relations r (R), whenever any two tuples t 1 and t 2 of r agree on the attributes  they also agree on the attributes . That is, t 1 [  ] = t 2 [  ]  t 1 [  ] = t 2 [  ]

4. Functional Dependencies between Attributes Superkeys and Keys  A superkey of a relation schema is a set of attributes that functionally determines all other attributes of the table  A set of attributes A is a key of a relation schema if A is a superkey and any proper subset of A is not a superkey  An attribute that is part of any key of the schema is called a prime attribute and an attribute that is not part of any key is a non-prime attribute

5. Functional Dependencies between Attributes Inferring Functional Dependencies  Inference process help us find functional dependencies in a relation, there are six rules: Reflexivity:If X  Y, then X  Y Augmentation:If X  Y, then XZ  YZ Transitivity:If X  Y and Y  Z, then X  Z Decomposition: If X  YZ, then X  Y Union:If X  Y and X  Z, then X  YZ Pseudo-transitivity:If X  Y and WX  Z, then WX  Z  Let W,X,Y, and Z be sets of functional dependencies and XY is the union of X and Y  The first three are known as Armstrong's Axioms which was used to derive the later rules as well

6. Functional Dependencies between Attributes Determining Keys from Functional Dependencies  Left side of a functional dependency is the determinant and the right side is known as the dependent  Formal Definition of the keys of a schema Suppose S is a schema that is specified as a set of attributes A and a set of functional dependencies F. Let F + be the closure of the functional dependencies defined for S. A set of attributes X is a key of S if and only if X  A- X is in F + and for each Y ⊂ X, Y  X is not in F +  If a functional dependency X  Z includes all of the attributes of the schema, then X is a superkey

7. Functional Dependencies between Attributes Functionally Dependencies Example #1  Functional Dependency defined on the attributes X and Y  Would read the relationship as “X determines Y” or “X  Y”  All the tuples in X that agree on X must also agree on Y

8. Functional Dependencies between Attributes Functionally Dependencies Example #2 Functional Dependencies in Schemas Employee: (EmpID, Name, Phone, Position)‏ EmpID  Name, Phone, Position Position  Phone

9. Normal Forms Normalization  Normalization is the process of transforming some objects into a structural form that satisfies some collection of rules  Used to reduce update anomalies and redundancy  A database schema is said to be in normal form if each of its relation schemas is in the normal form  A dependency X  Y is a full dependency if Y is not dependent on any subset of X  A dependency X  Y is a maximal dependency if there is no attribute A not in X or Y such that X ∪ Y {A}  A dependency X  Y is a trivial dependency if at least one attribute is part of both sides, that is, where X ∩ Y is not empty

10. Normal Forms First Normal Form: Atomic Attributes  The first normal form specifies that every attribute of a schema must takes its values from an atomic domain Not in First Normal Form First Normal Form

11. Normal Forms Second Normal Form: No Partial Key Dependencies  The second normal form is designed to eliminate functional dependencies that have part of a key on the left side  It is in second normal form if no non-prime attributes are partially dependent on any key of the schema and it is in (1NF)‏ Not in Second Normal Form Employee: (Emp_ID, Course_Title, Name, Dept_Name, Salary, Date_Completed)‏ Not fully functionally dependent on the primary key

12. Normal Forms Third Normal Form: No Transitive Dependencies  A schema is in third normal form if it has no transitive dependencies and it is in (2NF)‏  A transitive dependency of X  Y is a transitive dependency if there is a set of attributes Z that is not a subset of any key and both X  Z and Z  Y Must be dependent on the key this is not 3NF

13. Normal Forms Boyce-Codd Normal Form: No Non-Key Dependencies  It is similar to (3NF) but the table is in Boyce-Codd Normal Form if it has no non-key dependencies Not in Boyce-Codd Normal Form Supplier: (supplierID, supplierName, street, zipcode)‏ Functional Dependencies:  {supplierID, zipcode}  supplierName, street *Candidate Key  {supplierName, zipcode}  supplierID, street *Candidate Key  supplierName  supplierID  supplierID  supplierName  These are in (3NF) since they are part of the prime attributes but are not BCNF since supplierID  supplierName isn't satisfied  BCNF would be broken into  Supplier:(supplierID, street, zipcode)  SupplierName:(supplierID, supplierName)‏