1. Rensselaer Polytechnic Institute CSCI-4380 – Database Systems David Goldschmidt, Ph.D.

2.  A functional dependency on relation R is a logical expression of the form X  Y  X and Y are sets of attributes of R ▪ i.e. X = { A 1, A 2,..., A n } and Y = { B 1, B 2,..., B m } where n = m or n <> m  X  Y means that whenever any pair of tuples in R have the same values for attributes in X, then they must also have the same values for attributes in Y

3.  For each X  Y defined on relation R, it means that X functionally determines Y  Or more specifically, the attributes of X functionally determine the attributes of Y  More generally, a functional dependency adds meaning to attributes of R  In some cases, the occurrence of duplicate tuples does not make semantic sense

4.  For a given relation R, we look at the set of all functional dependencies to tell us what tuples we can (and should) store  We can also reason by applying simple inference rules to the tuples  e.g. transitivity, splitting/combining, etc.

5.  A constraint of any kind on relation R is said to be trivial if it holds for every instance of R  If Y  X, then X  Y is true for all relations  In other words, a trivial functional dependency has a right-hand side (Y) that is a subset of its left-hand side (X)  e.g. name artist  name  e.g. name  name What’s the point? We can remove trivial FDs! Reflexivity rule

6.  The functional dependency X  Y is equivalent to X  Z where attributes of Z are all those attributes of Y that are not also attributes of X  In other words, some of the attributes on the right-hand side (of X  Y) are also on the left (X)  We can simplify this by removing attributes from the right-hand side that also appear on the left

7.  Given functional dependency  X  Y  We can always add a set Z of attribute(s)  XZ  YZ  This is called augmentation

8.  Given functional dependency X  Y as  A 1, A 2,..., A n  B 1, B 2,..., B m  We can split it into multiple functional dependencies (singletons) as follows:  A 1, A 2,..., A n  B 1  A 1, A 2,..., A n  B 2 ...  A 1, A 2,..., A n  B m

9.  Given functional dependencies as follows:  A 1, A 2,..., A n  B 1  A 1, A 2,..., A n  B 2 ...  A 1, A 2,..., A n  B m  We can combine attributes on the right- hand side to form functional dependency  A 1, A 2,..., A n  B 1, B 2,..., B m

10.  Given functional dependencies  X  Y and Y  Z  We can unequivocally conclude that  X  Z  And if some attributes of Z are also attributes of X, we can eliminate them from the right-hand side (trivial-dependency rule)

11.  For a given relation R, we look at the set of functional dependencies to identify which attribute(s) imply all the rest  These attribute(s) form a key on R  Set K = { A 1, A 2,..., A n } is a key on R if:  K functionally determine all other attributes of R  No proper subset of K functionally determines all other attributes of R

12.  By definition, a key must be unique  A key K must functionally determine all other attributes of relation R  e.g. Student( id, name, address )  The key is the id attribute

13.  By definition, a key must be minimal  No proper subset of key K can functionally determine all other attributes of relation R  e.g. Student( id, name, address )  Even though id and name together might be unique, the id attribute is minimal

14.  A set of attributes that contains a key is called a superkey (a superset of a key)  The uniqueness constraint must be satisfied  The minimality constraint need not be satisfied  Every key is a superkey  e.g. Student( id, name, address )  Attribute id is both a key and a superkey  Attributes (id, name) form a superkey

15.  Model a US Census relation  Name, SSN, address, city, state, zip, area code, phone number, etc.  Use only a single relation  Describe functional dependencies  Identify keys and superkeys

16.  Given relation R with attributes A, B, C, D, E and A  BC, CD  E, BE  C  What does AE functionally determine (infer)?  In other words, AE  _____?

17.  Given a set of attributes X, the closure X+ is the set of attributes functionally determined by X  Given a relation R and a set F of functional dependencies, we need a way to find whether a functional dependency X  Y is true with respect to F

18.  Given relation R with attributes A, B, C, D, E and A  BC, CD  E, BE  C  AE  _____?  From reflexivity, AE+ = { A, E }  From A  BC, AE+ = { A, B, C, E }  No other rules are applicable or add to AE+  We conclude that AE  ABCE or simply AE  BC  Or AE  A, AE  B, AE  C, and AE  E

19.  Given a set F of functional dependencies, the closure X+ of a set of attributes X is determined by the following algorithm:  Initialize X+ to X  Repeat until X+ does not change: ▪ Find any unapplied functional dependency Y  Z in F such that Y  X+ ▪ Set X+ = X+  Z

20.  A set F of functional dependencies implies a functional dependency X  Y if Y  X+  In other words, if Y is in the closure of X, then functional dependency X  Y is true

21.  A key K for a given relation R is a minimal set of attributes A 1, A 2,..., A n such that closure {A 1, A 2,..., A n }+ is the set of all attributes of R  MusicGroup(name, artist, genre, dateformed, datefirstjoined)  name  genre dateformed  name artist  datefirstjoined  K must be (name, artist) because K+ = {name, artist, genre, dateformed, datefirstjoined}

22.  Review the US Census relation  What other functional dependencies can you infer?  Pick pairs of attributes (e.g. name and state) and identify the resulting closure  In other words, what is the set of attributes X+ functionally determined by set X (the pair)?