1. Functional Dependencies
Definition:
If two tuples agree on the attributes
A , A , … A 1 2 n
then they must also agree on the attributes
B , B , … B 1 2 m
Formally:
A , A , … A
B , B , … B 1 2 n 1 2 m
Motivating example for the study of functional dependencies:
Name Social Security Number Phone Number

2. Examples Product: name price, manufacturer Person: ssn name, age
Company: name stock price, president
Key of a relation is a set of attributes that:
- functionally determines all the attributes of the relation
- none of its subsets determines all the attributes.
Superkey: a set of attributes that contains a key.

3. Finding the Attributes of a Relation
Given a relation constructed from an E/R diagram, what is its key?
Rules:
1. If the relation comes from an entity set,
the key of the relation is the set of attributes which is the
key of the entity set.
Person
name
ssn
address

4. Rules for Binary Relationships
name
buys
Person
Product
price
name
ssn
Several cases are possible for a binary relationship E1 - E2:
1. Many-many: the key includes the of E1 together with the
key of E2.
What happens for:
2. Many-one:
3. One-one:

5. Rules for Multiway Relationships
None, really.
Except: if there is an arrow from the relationship to E, then
we don’t need the key of E as part of the relation key.
Product
Purchase
Store
Payment Method
Person

6. Some Properties of FD’s
A , A , … A
B , B , … B
Is equivalent to 1 2 n 1 2 m
A , A , … A B 1 2 n 1
Splitting rule
and
Combing rule
A , A , … A B 1 2 n 2 …
A , A , … A B 1 2 n m
A , A , … A A
Always holds. 1 2 n i

7. Comparing Functional Dependencies
Entailment: a set of functional dependencies S1 entails a set S2 if:
any database that satisfies S1 much also satisfy S2.
Example: A B, B C
entails
A C
Equivalence: two sets of FD’s are equivalent if each entails the
other.
{A B, B C } is equivalent to {A B, A C, B C}

8. Closure of a set of Attributes
Given a set of attributes A and a set of dependencies C,
we want to find all the other attributes that are functionally
determined by A.
In other words, we want to find the maximal set of attributes B,
such that for every B in B,
C entails A B.

9. Closure Algorithm Start with Closure=A.
Until closure doesn’t change do:
if is in C, and
B is not in Closure
then
add B to closure.
A , A , … A B 1 2 n
A , A , … A
Are all in the closure, and 1 2 n

10. Example
A B C
A D E
B D
A F B
Closure of {A,B}:
Closure of {A, F}: