1. The Relational Data Model and Relational Database Constraints

2. Outline Relational Model Concepts Characteristics of Relations
Relational Model Constraints
Key Constraints
Entity Integrity Constraints
Referential Integrity Constraints
Update Operations on Relations .

3. Relational Model Concepts
Basis of the model
The relational model of data is based on the concept of a Relation.
A Relation is a mathematical concept based on the ideas of sets.
The strength of the relational approach to data management comes from the formal foundation provided by the theory of relations.
We review the essentials of the relational approach in this chapter.

4. Formal Definitions Domain Attribute A set of atomic values
Given with a name, data type, and format
Attribute
The name of a role played by some domain D
D is called the domain of Ai: dom(Ai)

5. Formal Definitions Relation schema N-tuple
Defined over attributes A1, A2, …, An
Denoted R(A1,A2, …,An)
n is the degree (arity) of the relation
N-tuple
An ordered list of n values
Denoted t = <v1,v2,…,vn>
vi is an element of some domain Ai, denoted t[Ai]

6. Formal Definitions Relation
A relation (or relation state) r of R(A1,A2,…,An) is a set of n-tuples r = {t1,t2,…,tm}.
Each value vi is an element of dom(Ai).
Denoted r(R)

7. Example CUSTOMER (Cust-id, Cust-name, Address, Phone#)
CUSTOMER is a relation defined over the four attributes Cust-id, Cust-name, Address, Phone#,
Each attribute has a domain or a set of valid values. E.g., the domain of Cust-id is 6 digit numbers.
<632895, "John Smith", "101 Main St. Atlanta, GA ", "(404) ">, for instance, is a tuple belonging to the CUSTOMER relation.
A relation (table) may be regarded as a set of tuples (rows).
Attributes of the relation correspond to columns in the table.

8. Figure 5.1 The attributes and tuples of a relation STUDENT.

9. Mathematical Relation
Relation r(R) where R(A1, A2, ..., An)
Mathematical relation of degree n on the domains dom(A1), dom(A2), …, dom(An).
A subset of the Cartesian product of the domains
r(R)  dom(A1) × dom(A2) × .... × dom(An)
The Cartesian product specifies all possible combinations of values from the underlying domains:
|dom(A1)| × |dom(A2)| × .... × |dom(An)|
r(R) is a set of valid tuples.

10. Example r(R): a specific "value" or population of R.
Let S1 = {0,1}, S2 = {a,b,c}.
Let r(R)  S1 × S2
For example: r(R) = {<0.a>, <0,b>, <1,c>}
r(R): a specific "value" or population of R.
R is also called the intension of a relation
r is also called the extension of a relation

11. Definition Summary Informal Terms Formal Terms Table Relation
Column Attribute/Domain
Row Tuple
Values in a column Domain
Table Definition Relation Schema
Populated Table Extension

12. Characteristics of Relations
Ordering of tuples
Tuple ordering is not part of a relation definition
Tuples are not considered to be ordered, even though they appear to be in the tabular form.
Logical orders can be specified, only to facilitate functions such as searching.
Ordering of attributes
Attributes in R(A1, A2, ..., An) and the values in t=<v1,v2, ..., vn> are considered to be ordered.
Alternative definition of relation is possible

13. Figure 5. 2 The relation STUDENT from Figure 5
Figure 5.2 The relation STUDENT from Figure 5.1, with a different order of tuples.

14. Characteristics of Relations
Values in a tuple:
All values are considered atomic (indivisible).
Multivalued attributes must be represented by separate relations.
Composite attributes are represented only by their simple component attributes.
A special null value is used to represent values that are unknown or inapplicable to certain tuples.

15. Characteristics of Relations
Interpretation of a relation
Relations represent facts about entities and relationships
How to converse EER constructs to relations
Notation
Relation name: Q, R, S
Relation states: q, r, s
Tuples: t, u, v
t[Ai] = vi (the value of attribute Ai for tuple t).
t[Au,Av,...,Aw] refers to the subtuple of t containing the values of attributes Au, Av, ..., Aw, respectively.

16. Relational Model Constraints
Categories
Inherent model-based constraints
e.g. no duplicate tuples
Schema-based constraints
Expressed in the schema
Application-based constraints
Must be checked within application programs
Data dependencies
Used in the design of a relational database

17. Relational Model Constraints
Conditions that must hold on all valid relation instances.
Types of constraints:
Domain constraints
Key constraints
Entity integrity constraints
Referential integrity constraints

18. Domain Constraints Domain constraints
The value of each attribute A must be an atomic value from dom(A)
Typical data types: integers, real numbers, characters, booleans, strings

19. Key Constraints Superkey of R: Key of R:
A set of attributes SK of R, that specifies a uniqueness constraint
No two tuples in any valid relation instance r(R) will have the same value for SK. That is, for any distinct tuples t1 and t2 in r(R), t1[SK]  t2[SK].
Key of R:
A "minimal" superkey; that is, a superkey K such that removal of any attribute from K results in a set of attributes that is not a superkey.

20. Key Constraints Example: The CAR relation schema:
CAR(State, Reg#, SerialNo, Make, Model, Year)
has two keys Key1 = {State, Reg#}, Key2 = {SerialNo}, which are also superkeys. {SerialNo, Make} is a superkey but not a key. 
If a relation has several candidate keys, one is chosen arbitrarily to be the primary key. The primary key attributes are underlined.

21. Figure 5.4 The CAR relation with two candidate keys:
LicenseNumber and EngineSerialNumber.

22. t[PK]  null for any tuple t in r(R)
Entity Integrity
Relational Database Schema:
A set S of relation schemas that belong to the same database. S is the name of the database.
S = {R1, R2, ..., Rn}
Entity Integrity:
The primary key attributes PK of each relation schema R in S cannot have null values in any tuple of r(R). This is because primary key values are used to identify the individual tuples.
t[PK]  null for any tuple t in r(R)

23. Figure 5.5 Schema diagram for the COMPANY relational database schema; the primary keys are underlined.

24. Figure 5.6 One possible relational database state corresponding to the COMPANY schema.

26. Referential Integrity
A constraint involving two relations (the previous constraints involve a single relation).
Used to specify a relationship among tuples in two relations: the referencing relation and the referenced relation.
Tuples in the referencing relation R1 have attributes FK (called foreign key attributes) that reference the primary key attributes PK of the referenced relation R2. A tuple t1 in R1 is said to reference a tuple t2 in R2 if t1[FK] = t2[PK].
A referential integrity constraint can be displayed in a relational database schema as a directed arc from R1. FK to R2.

27. Figure 5.7 Referential integrity constraints displayed on the COMPANY relational database schema diagram.

28. Update Operations Operations Constraints
INSERT, DELETE, MODIFY form a new relation.
Applying algebraic operators (relational algebra)
Defining the new relation (relational calculus)
Constraints
Update operations should not violate integrity constraints
Several update operations may have to be grouped together.
Updates may propagate to cause other updates automatically. This may be necessary to maintain integrity constraints.

29. Constraint Violations
In case of integrity violation:
Cancel the operation that causes the violation (REJECT option)
Perform the operation but inform the user of the violation
Trigger additional updates so the violation is corrected (CASCADE option, SET NULL option)
Execute a user-specified error-correction routine