1. Lecture 08: E/R Diagrams and Functional Dependencies
Friday, October 15, 2004

2. Outline Finish E/R diagrams (Chapter 2) The relational data model: 3.1
And E/R diagrams to relations (3.2, 3.3)
The relational data model: 3.1
Functional dependencies: 3.4

3. Constraints in E/R Diagrams
Finding constraints is part of the modeling process.
Commonly used constraints:
Keys: social security number uniquely identifies a person.
Single-value constraints: a person can have only one father.
Referential integrity constraints: if you work for a company, it
must exist in the database.
Other constraints: peoples’ ages are between 0 and 150.

4. Keys in E/R Diagrams name category Underline: price Product
No formal way
to specify multiple
keys in E/R diagrams
Person
name
ssn
address

5. Single Value Constraints
makes
v. s.
makes

6. Referential Integrity Constraints
makes
Product
Company
Each product made by at most one company.
Some products made by no company
makes
Product
Company
Each product made by exactly one company.

7. Other Constraints
makes
<100
Product
Company
What does this mean ?

8. Weak Entity Sets Entity sets are weak when their key comes from other
classes to which they are related.
affiliation
Team
University
sport
number
name
Notice: we encountered this when converting multiway relationships to binary relationships (last lecture)

9. Handling Weak Entity Sets
affiliation
Team
University
sport
number
name
Convert to a relational schema (in class)

10. The Relational Data Model
Schema
Physical
storage
Data
Modeling
Have seen this in SQL
Complex
file organization
and index
structures.
E/R diagrams
Tables:
column names: attributes
rows: tuples
Have seen this too
Discuss next

11. Terminology Table name or relation name Attribute names Products:
Name Price Category Manufacturer
gizmo $ gadgets GizmoWorks
Power gizmo $ gadgets GizmoWorks
SingleTouch $ photography Canon
MultiTouch $ household Hitachi
Tuples or rows or records

12. Schemas Relational Schema: Relation name plus attribute names
E.g. Product(Name, Price, Category, Manufacturer)
In practice we add the domain for each attribute
Database Schema
Set of relational schemas
E.g. Product(Name, Price, Category, Manufacturer),
Company(Name, Address, Phone),
This is all mathematics, not to be confused with SQL tables !

13. Instances
Relational schema = R(A1,…,Ak): Instance = relation with k attributes (of “type” R)
values of corresponding domains
Database schema = R1(…), R2(…), …, Rn(…) Instance = n relations, of types R1, R2, ..., Rn

14. Example Relational schema:Product(Name, Price, Category, Manufacturer)
Instance:
Name Price Category Manufacturer
gizmo $ gadgets GizmoWorks
Power gizmo $ gadgets GizmoWorks
SingleTouch $ photography Canon
MultiTouch $ household Hitachi

15. First Normal Form (1NF)
A database schema is in First Normal Form if all tables are flat
Student
Student
Name
GPA
Alice
3.8
Bob
3.7
Carol
3.9
Name
GPA
Courses
Alice
3.8
Bob
3.7
Carol
3.9
Math DB OS
Takes
Course
Student
Course
Alice
Math
Carol DB
Bob OS
Course
Math DB OS DB OS
May need to add keys
Math OS

16. Functional Dependencies
A form of constraint
hence, part of the schema
Finding them is part of the database design
Also used in normalizing the relations

17. Functional Dependencies
Warning: this is the most abstract, and “hardest” part of the course.
This chapter in the book (3.4) is not very good
COME TO CLASS, PAY ATTENTION, READ THE LECTURE NOTES !

18. Functional Dependencies
Definition:
If two tuples agree on the attributes
A1, A2, …, An
then they must also agree on the attributes
B1, B2, …, Bm
Formally:
A1, A2, …, An  B1, B2, …, Bm

19. Typical Examples of FDs
Product: name  price, manufacturer
Person: ssn  name, age
zip  city
city, state  zip

20. Examples EmpID Name Phone Position E0045 Smith 1234 Clerk E1847 John
Formally, an FD holds, or does not hold on an instance:
EmpID
Name
Phone
Position
E0045
Smith
1234
Clerk
E1847
John
9876
Salesrep
E1111
E9999
Mary
Lawyer
EmpID  Name, Phone, Position
Position  Phone
but not Phone  Position

21. In General To check A  B, erase all other columns
check if the remaining relation is many-one (called functional in mathematics)
Note: this is the mathematical definition of a function. Book is wrong.

22. Example EmpID Name Phone Position E0045 Smith 1234 Clerk E1847 John
9876
Salesrep
E1111
Smith
9876
Salesrep
E9999
Mary
1234
Lawyer

23. Example FD’s are constraints: On some instances they hold name  color
On others they don’t
name  color
category  department
color, category  price
name
category
color
department
price
Gizmo
Gadget
Green
Toys 49
Tweaker 99
No: color,category  price doesn’t hold
Does this instance satisfy all the FDs ?

24. Example name  color category  department color, category  price
Gizmo
Gadget
Green
Toys 49
Tweaker
Black 99
Stationary
Office-supp. 59
yes
What about this one ?

25. An Interesting Observation
name  color
category  department
color, category  price
If all these FDs are true:
Then this FD also holds:
name, category  price
Why ??

26. Inference Rules for FD’s
A1, A2, …, An  B1, B2, …, Bm
Splitting rule
and
Combing rule
Is equivalent to A1
... Am B1 Bm
A1, A2, …, An  B1
A1, A2, …, An  B2
A1, A2, …, An  Bm

27. Inference Rules for FD’s (continued)
Trivial Rule
A1, A2, …, An  Ai
where i = 1, 2, ..., n A1 … Am
Why ?

28. Inference Rules for FD’s (continued)
Transitive Closure Rule If
A1, A2, …, An  B1, B2, …, Bm
and
B1, B2, …, Bm  C1, C2, …, Cp
then
A1, A2, …, An  C1, C2, …, Cp
Why ?

29. A1 … Am B1 Bm C1
... Cp

30. Example (continued) Which Rule did we apply ? Inferred FD
1. name  color
2. category  department
3. color, category  price
Start from the following FDs:
Infer the following FDs:
Inferred FD
Which Rule did we apply ?
4. name, category  name
5. name, category  color
6. name, category  category
7. name, category  color, category
8. name, category  price

31. Example (continued) 1. name  color 2. category  department
3. color, category  price
Answers:
Inferred FD
Which Rule did we apply ?
4. name, category  name
Trivial rule
5. name, category  color
Transitivity on 4, 1
6. name, category  category
7. name, category  color, category
Split/combine on 5, 6
8. name, category  price
Transitivity on 3, 7
THIS IS TOO HARD ! Let’s see an easier way.

32. Closure of a set of Attributes
Given a set of attributes A1, …, An
The closure, {A1, …, An}+ , is the set of attributes B s.t. A1, …, An  B
Example:
name  color
category  department
color, category  price
Closures: name+ = {name, color} {name, category}+ = {name, category, color, department, price} color+ = {color}

33. Closure Algorithm Start with X={A1, …, An}.
Repeat until X doesn’t change do:
if B1, …, Bn  C is a FD and
B1, …, Bn are all in X
then add C to X.
Example:
name  color
category  department
color, category  price
{name, category}+ = { }
name, category, color, department, price
Hence:
name, category  color, department, price

34. Another Example Enrollment(student, major, course, room, time)
course  time
What else can we infer ? [in class, or at home]

35. Example In class: A, B  C R(A,B,C,D,E,F) A, D  E B  D A, F  B
Compute {A,B}+ X = {A, B, }
Compute {A, F}+ X = {A, F, }

36. Using Closure to Infer ALL FDs
Example:
A, B  C A, D  B
B  D
Step 1: Compute X+, for every X:
A+ = A, B+ = BD, C+ = C, D+ = D
AB+ = ABCD, AC+ = AC, AD+ = ABCD
ABC+ = ABD+ = ACD+ = ABCD (no need to compute– why ?)
BCD+ = BCD, ABCD+ = ABCD
Step 2: Enumerate all FD’s X  Y, s.t. Y  X+ and XY = :
AB  CD, ADBC, ABC  D, ABD  C, ACD  B