NOTES - PART III

The Relational Data Model

(Based on Chapter 7 in

Fundamentals of Database Systems by Elmasri and Navathe, Ed. 3)

1 Relational Model Concepts

2 Characteristics of Relations

3 Relational Integrity Constraints

3.1 Key Constraints

3.2 Entity Integrity Constraints

3.3 Referential Integrity Constraints

4 Update Operations on Relations

5 Relational Algebra Operations

5.1 SELECT s and PROJECT P

5.2 Set Operations

5.3 JOIN Operations

5.4 Additional Relational Operations

1. 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.

__________________________________________________________

1. The relational model is covered in Chapter 7 of the book Fundamentals of Database Systems, by Elmasri and S.B. Navathe, Ed. III, 2000.

2. The model was first proposed by Dr. E.F. Codd of IBM in 1970 in the following paper - "A Relational Model for Large Shared Data Banks," Communications of the ACM, June 1970.

INFORMAL DEFINITIONS

• RELATION: A table of values

• A relation may be thought of as a set of rows.

• A relation may alternately be though of as a set of columns.

• Each row of the relation may be given an identifier.

• Each column typically is called by its column name or column header or attribute name.

FORMAL DEFINITIONS

• A Relation may be defined in multiple ways.

• The Schema of a Relation:

R (A₁, A₂, .....A_n)

Relation R is defined over attributes A_1, A_2, .....A_n

_{For Example -}

_{CUSTOMER (Cust-id, Cust-name, Address, Phone#)}

Here, CUSTOMER is a relation defined over the four attributes Cust-id, Cust-name, Address, Phone#, each of which has a domain or a set of valid values. For example, the domain of Cust-id is 6 digit numbers.

• A tuple is an ordered set of values

• Each value is derived from an appropriate domain.

• Each row in the CUSTOMER table may be called as a tuple in the table and would consist of four values.

<632895, "John Smith", "101 Main St. Atlanta, GA 30332", "(404) 894-2000"> is a triple belonging to the CUSTOMER relation.

• A relation may be regarded as a set of tuples (rows).

• Columns in a table are also called as attributes of the relation.

FORMAL DEFINITIONS (contd.)

• The relation is formed over the cartesian product of the sets; each set has values from a domain; that domain is used in a specific role which is conveyed by the attribute name.

• For example, attribute Cust-name is defined over the domain of strings of 25 characters. The role these strings play in the CUSTOMER relation is that of the name of customers.

• Formally,

Given R(A₁, A₂, .........., A_n)

r(R) subset-of dom ₍A_{1) X dom (}A₂) X ....X dom(A_n)

R: schema of the relation

r of 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

Let S1 = {0,1}

Let S2 = {a,b,c}

Let R subset-of S1 X S2

for example: r(R) = {<0,a> , <0,b> , <1,c> }

DEFINITION SUMMARY

Informal Terms Formal Terms

Table Relation

Column Attribute/Domain

Row Tuple

Values in a column Domain

Table Definition Schema of Relation

Populated Table Extension

Notes:

Whereas languages like SQL use the informal terms of TABLE (e.g. CREATE TABLE), COLUMN (e.g. SYSCOLUMN variable), the relational database textbooks present the model and operations on it using the formal terms.

2 Characteristics of Relations

Ordering of tuples in a relation r(R): The tuples are not considered to be ordered, even though they appear to be in the tabular form.

Ordering of attributes in a relation schema R (and of values within each tuple): We will consider the attributes in R(A₁, A₂, ..., A_n) and the values in t=<v₁, v₂, ..., v_n> to be ordered .

(However, a more general alternative definition of relation does not require this ordering).

Values in a tuple: All values are considered atomic (indivisible). A special null value is used to represent values that are unknown or inapplicable to certain tuples.

Notation:

- We refer to component values of a tuple t by t[A_i] = v_i (the value of attribute A_i for tuple t).

Similarly, t[A_u, A_v, ..., A_w] refers to the subtuple of t containing the values of attributes A_u, A_v, ..., A_w, respectively.

3 Relational Integrity Constraints

Constraints are conditions that must hold on all valid relation instances. There are three main types of constraints:

Key constraints, entity integrity constraints, and referential integrity constraints

3.1 Key Constraints

Superkey of R: A set of attributes SK of R such that 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.

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.

3.2 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, ..., R_n}

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)

Note: Other attributes of R may be similarly constrained to disallow null values, even though they are not members of the primary key.

3.3 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.

4 Update Operations on Relations

- INSERT a tuple.

- DELETE a tuple.

- MODIFY a tuple.

- Integrity constraints should not be violated by the update operations.

- 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.

- In case of integrity violation, several actions can be taken:

- cancel the operation that causes the violation (REJECT optiom)

- 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

5 The Relational Algebra

- Operations to manipulate relations.

- Used to specify retrieval requests (queries).

- Query result is in the form of a relation.

Relational Operations:

5.1 SELECT s and PROJECT P operations.

5.2 Set operations: These include UNION U, INTERSECTION | |, DIFFERENCE -, CARTESIAN PRODUCT X.

5.3 JOIN operations X.

5.4 Other relational operations: DIVISION, OUTER JOIN, AGGREGATE FUNCTIONS.

5.1 SELECT s and PROJECT P

SELECT operation (denoted bys ):

- Selects the tuples (rows) from a relation R that satisfy a certain selection condition c

- Form of the operation: s _c(R)

- The condition c is an arbitrary Boolean expression on the attributes of R

- Resulting relation has the same attributes as R

- Resulting relation includes each tuple in r(R) whose attribute values satisfy the condition c

Examples:

s _DNO=4(EMPLOYEE)

s _SALARY>30000(EMPLOYEE)

s_{(DNO=4 AND SALARY>25000) OR DNO=5}(EMPLOYEE)

PROJECT operation (denoted byP ):

- Keeps only certain attributes (columns) from a relation R specified in an attribute list L

- Form of operation: P _L(R)

- Resulting relation has only those attributes of R specified in L

Example: P _{FNAME,LNAME,SALARY}(EMPLOYEE)

- The PROJECT operation eliminates duplicate tuples in the resulting relation so that it remains a mathematical set (no duplicate elements)

Example: P _SEX,SALARY(EMPLOYEE)

If several male employees have salary 30000, only a single tuple <M, 30000> is kept in the resulting relation.

Duplicate tuples are eliminated by the P operation.

Sequences of operations:

- Several operations can be combined to form a relational algebra expression (query)

Example: Retrieve the names and salaries of employees who work in department 4:

P _{FNAME,LNAME,SALARY} (s _DNO=4(EMPLOYEE) )

- Alternatively, we specify explicit intermediate relations for each step:

DEPT4_EMPS <-s _DNO=4(EMPLOYEE)

R <-P _{FNAME,LNAME,SALARY}(DEPT4_EMPS)

- Attributes can optionally be renamed in the resulting left-hand-side relation (this may be required for some operations that will be presented later):

DEPT4_EMPS <-s _DNO=4(EMPLOYEE)

R(FIRSTNAME,LASTNAME,SALARY) <-

P _{FNAME,LNAME,SALARY}(DEPT4_EMPS)

5.2 Set Operations

- Binary operations from mathematical set theory:

UNION: R₁ U R₂,

INTERSECTION: R₁ | | R₂,

SET DIFFERENCE: R₁ - R₂,

CARTESIAN PRODUCT: R₁ X R₂.

- For U, | |, -, the operand relations R₁(A₁, A₂, ..., A_n) and R₂(B₁, B₂, ..., B_n) must have the same number of attributes, and the domains of corresponding attributes must be compatible; that is, dom(A_i)=dom(B_i) for i=1, 2, ..., n. This condition is called union compatibility.

- The resulting relation for U, | |, or - has the same attribute names as the first operand relation R₁ (by convention).

CARTESIAN PRODUCT

R(A₁, A₂, ..., A_m, B₁, B₂, ..., B_n) <-

R₁(A₁, A₂, ..., A_m) X R₂ (B₁, B₂, ..., B_n)

- A tuple t exists in R for each combination of tuples t₁ from R₁ and t₂ from R₂ such that:

t[A₁, A₂, ..., A_m]=t₁ and t[B₁, B₂, ..., B_n]=t₂

- If R₁ has n₁ tuples and R₂ has n₂ tuples, then R will have n₁*n₂ tuples.

- CARTESIAN PRODUCT is a meaningless operation on its own. It can combine related tuples from two relations if followed by the appropriate SELECT operation .

Example: Combine each DEPARTMENT tuple with the EMPLOYEE tuple of the manager.

DEP_EMP <-DEPARTMENT X EMPLOYEE

DEPT_MANAGER <-s _MGRSSN=SSN(DEP_EMP)

5.3 JOIN Operations

THETA JOIN: Similar to a CARTESIAN PRODUCT followed by a SELECT. The condition c is called a join condition.

R(A₁, A₂, ..., A_m, B₁, B₂, ..., B_n) <-

R₁(A₁, A₂, ..., A_m) X _c R₂ (B₁, B₂, ..., B_n)

EQUIJOIN: The join condition c includes one or more equality comparisons involving attributes from R₁ and R₂. That is, c is of the form:

(A_i=B_j) AND ... AND (A_h=B_k); 1<i,h<m, 1<j,k<n

In the above EQUIJOIN operation:

A_i, ..., A_h are called the join attributes of R₁

B_j, ..., B_k are called the join attributes of R₂

Example of using EQUIJOIN:

Retrieve each DEPARTMENT's name and its manager's name:

T <-DEPARTMENT X_MGRSSN=SSN EMPLOYEE

RESULT <-P _{DNAME,FNAME,LNAME}(T)

NATURAL JOIN (*):

In an EQUIJOIN R <- R₁ X _c R₂, the join attribute of R₂ appear redundantly in the result relation R. In a NATURAL JOIN, the redundant join attributes of R₂ are eliminated from R. The equality condition is implied and need not be specified.

R <- R₁ *_{(join attributes of R1),(join attributes of R2)} R₂

Example: Retrieve each EMPLOYEE's name and the name of the DEPARTMENT he/she works for:

T<- EMPLOYEE *_{(DNO),(DNUMBER)} DEPARTMENT

RESULT <-P _{FNAME,LNAME,DNAME}(T)

If the join attributes have the same names in both relations, they need not be specified and we can write R <- R₁ * R₂.

Example: Retrieve each EMPLOYEE's name and the name of his/her SUPERVISOR:

SUPERVISOR(SUPERSSN,SFN,SLN)<-

P _{SSN,FNAME,LNAME}(EMPLOYEE)

T<-EMPLOYEE * SUPERVISOR

RESULT <-P _{FNAME,LNAME,SFN,SLN}(T)

Note: In the original definition of NATURAL JOIN, the join attributes were required to have the same names in both relations.

There can be a more than one set of join attributes with a different meaning between the same two relations. For example:

JOIN ATTRIBUTES RELATIONSHIP

EMPLOYEE.SSN= EMPLOYEE manages

DEPARTMENT.MGRSSN the DEPARTMENT

EMPLOYEE.DNO= EMPLOYEE works for

DEPARTMENT.DNUMBER the DEPARTMENT

Example: Retrieve each EMPLOYEE's name and the name of the DEPARTMENT he/she works for:

T<-EMPLOYEE X_DNO=DNUMBERDEPARTMENT

RESULT <-P _{FNAME,LNAME,DNAME}(T)

A relation can have a set of join attributes to join it with itself :

JOIN ATTRIBUTES RELATIONSHIP

EMPLOYEE(1).SUPERSSN= EMPLOYEE(2) supervises

EMPLOYEE(2).SSN EMPLOYEE(1)

- One can think of this as joining two distinct copies of the relation, although only one relation actually exists

- In this case, renaming can be useful

Example: Retrieve each EMPLOYEE's name and the name of his/her SUPERVISOR:

SUPERVISOR(SSSN,SFN,SLN)<-

P _{SSN,FNAME,LNAME}(EMPLOYEE)

T<-EMPLOYEE X_{SUPERSSN=SSSN}SUPERVISOR

RESULT <-P _{FNAME,LNAME,SFN,SLN}(T)

Complete Set of Relational Algebra Operations:

- All the operations discussed so far can be described as a sequence of only the operations SELECT, PROJECT, UNION, SET DIFFERENCE, and CARTESIAN PRODUCT.

- Hence, the set {s ,P , U, - , X } is called a complete set of relational algebra operations. Any query language equivalent to these operations is called relationally complete.

- For database applications, additional operations are needed that were not part of the original relational algebra. These include:

1. Aggregate functions and grouping.

2. OUTER JOIN and OUTER UNION.

5.4 Additional Relational Operations

AGGREGATE FUNCTIONS

- Functions such as SUM, COUNT, AVERAGE, MIN, MAX are often applied to sets of values or sets of tuples in database applications

_{<grouping attributes>} F_{<function list>} (R)

- The grouping attributes are optional

Example 1: Retrieve the average salary of all employees (no grouping needed):

R(AVGSAL) <- F _{AVERAGE SALARY} (EMPLOYEE)

Example 2: For each department, retrieve the department number, the number of employees, and the average salary (in the department):

R(DNO,NUMEMPS,AVGSAL) <-

_DNO F _{COUNT SSN, AVERAGE SALARY} (EMPLOYEE)

DNO is called the grouping attribute in the above example

OUTER JOIN

- In a regular EQUIJOIN or NATURAL JOIN operation, tuples in R₁ or R₂ that do not have matching tuples in the other relation do not appear in the result

- Some queries require all tuples in R₁ (or R₂ or both) to appear in the result

- When no matching tuples are found, nulls are placed for the missing attributes

- LEFT OUTER JOIN: R₁ X R₂ lets every tuple in R₁ appear in the result

- RIGHT OUTER JOIN: R₁ X R₂ lets every tuple in R₂ appear in the result

- FULL OUTER JOIN: R₁ X R₂ lets every tuple in R₁ or R₂ appear in the result