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The Relational Data Model, Relational Constraints,

and The Relational Algebra

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Contents• Relational Model Concepts• Relational Constraints and Relational Database

Schemas• Basic Relational Algebra Operations• Additional Relational Operations

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Relational Model Concepts• The Relational model represents the database as a collection of

relations• Relation is thought of as “table” of values, where each row

represents a collection of related data values.• Table name and Column names are used to help in interpreting

the meaning of the values in each row.• In the formal relational model: row – tuple, column header –

attribute, table – relation, type of values that can appear in each column – domain.

Domains: • A domain D is a set of atomic values.• A domain is given a name, data type and format. Units can also be

given to a domain.• R (A1, A2, A3…, An)• D = dom (Ai)• R – name of the relation• n – degree of a relation

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ExampleSTUDENT (Name, IRD, HomePhone, Address, OfficePhone, Age, GPA)• What is the relation name?• What is the degree of the relation?• What are the attributes?• List each dom(Ai).

• A relation r (or relation state) of the relation schema R(A1, A2, . . ., An), also denoted by r(R), is a set of n tuples r = {t1, t2, . . ., tk}.

• Each n-tuple t is an ordered list of n values t = <v1, v2, . . ., vn>, where each value vi is an element of dom(Ai) or is a special null value.

• Possible for several attributes to have the same domain. The attributes indicate different roles or interpretations for the domain.

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Characteristics of Relations• Ordering of tuples in a relation

» No• Ordering of values within a tuple

» Yes• Values in the tuples

» is an atomic value» composite and multivalued attributes are not allowed

• How to represent multi-valued attributes?» by separate relation

• How to represent composite attributes?» only by their simple component attributes.

• NULL» Value unknown» May not apply» exists but not available

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Relational Model Notations• Q, R, S denote relation names• q, r, s denote relation states• t, u, v denote tuples• STUDENT indicates the current set of tuples• STUDENT(Name, IRD, . . .) refers only to the relation schema• An attribute can be qualified with a relation name, such as R.A

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Relational Constraints and Relational Database Schemas• Domain Constraints• Key Constraints and Constraints on Null• Relational Databases and Relational Database Schemas• Entity Integrity, Referential Integrity and Foreign Keys

Domain Constraints• Domain constraints specify that the value of each attribute ‘A’

must be an atomic value from the domain.• Data types are usually associated with domains• Examples• Subrange

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Key Constraints and Constraints on Null• No two tuples can have the same values for all their attributes.• Superkey - a set of attributes such that for any two distinct tuples

t1 and t2» t1[SK] != t2[SK]» Any set of attributes satisfying the above is a superkey

• The superkey specifies a uniqueness constraint on the tuples• A key K of a relation schema R is a superkey of R with the

additional property that removing any of the attributes from K leaves a set of attributes that is not a superkey.

• A relation schema may have more than one key. Each of the keys is called a candidate key.

• Typically one candidate key is designated the primary key.

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Relational Databases and Relational Database Schemas

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Entity Integrity• The primary key is used to identify individual tuples.• The entity integrity constraint states that no primary key value can

be null.• Key constraints and entity constraints are specified on individual

relations.

Referential Integrity• The referential integrity constraint is specified between two

relations. It is used to maintain consistency among tuples of the two relations.

• Informally, it states that a tuple in one relation that refers to another relation must refer to an existing tuple in that relation.

• Example

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Foreign Keys

• A set of attributes FK in relation schema R1 is a foreign key of R1 that references relation R2 if it satisfies these conditions:» The attributes of FK have the same domain(s) as the primary key attributes PK of R2» A value of FK in a tuple t1 of the current state r1(R1) either occurs as a value of PK for some tuple t2 in the current state r2(R2) or is null.

• The attributes FK are said to refer to the relation R2.• From (2), we have t1[FK] = t2[PK].

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Basic Relational Algebra Operations• Need a set of operations• Relational algebra• Retrieval can be from one relation or more than one relation• Result is a relation• Relational algebra expression – sequence of relational algebra

operations• Operations divided into two groups

» Set operations from mathematical set theory: UNION, INTERSECTION, SET DIFFERENCE and CARTESIAN PRODUCT.

» Operations developed for relational databases: SELECT, PROJECT, JOIN

SELECT:• σ<selection condition>(R)

<attr name> <comparison op> <constant value><attr name> <comparison op> <attr name>

• σDNO =4(EMPLOYEE)• σSALARY >30000(EMPLOYEE)• σ(DNO =4 AND SALARY>25000) OR DNO=5 AND SALARY>30000)(EMPLOYEE)

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• SELECT is commutativeσ<cond1>(σ<cond2>(R)) = σ<cond2>(σ<cond1>(R))

• A cascade of SELECTs can be combined into a single SELECTσ<cond1>(σ<cond2>(. . .(σ<condn>(R)) = σ<cond1> AND <cond2> AND . . AND <condn>(R)

PROJECT:• Selects some attributes from a relation• Π<attribute list>(R)• The result is a relation having only the attributes in the attribute

list.• Order of attributes is the <attribute list> order.• The degree of the relation is the number of attributes in the list.• Duplicates are removed.• Π<list1>(Π<list2>(R)) = Π<list1>(R)

» As long as <list2> contains the attributes in <list1>• Commutativity does not hold for PROJECT.• Example:

ΠLNAME, FNAME, SALARY(EMPLOYEE)ΠSEX, SALARY(EMPLOYEE)

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Sequences of Operations• Operations can be

» Nested» Applied one (or a few) at a time, creating intermediate

results• Example

» ΠFNAME, LNAME, SALARY (σDNO=5(EMPLOYEE))or

» DEP5_EMPS ← σDNO=5(EMPLOYEE)» RESULT ← ΠFNAME, LNAME, SALARY(DEP5_EMPS)

RENAME:• ρS (R)• ρ(B1, B2, . . ., Bn) (R)• ρS(B1, B2, . . ., Bn) (R)• ρ (rho) – RENAME Operator• S is the new relation name.• B1, B2, . . ., Bn are the new attribute names.

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Set Theoretic Operations• UNION

» R ∪ S• Includes all tuples that are in R or in S or in both• INTERSECTION

» R ∩ S• Includes every tuple that is simultaneously in both R and S• SET DIFFERENCE

» R - S» Includes all tuples that are in R but not is S

• UNION and INTERSECTION are commutative» R ∪ S = S ∪ R» R ∩ S = S ∩ R

• UNION and INTERSECTION are associative» R ∪ (S ∪ T) = (R ∪ S) ∪ T» R ∩ (S ∩ T) = (R ∩ S) ∩ T

• SET DIFFERENCE is not commutative» R - S != S - R

• Must be union compatible » Have the same degree n» dom(Ai) = dom(Bi) for all i

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DEPT3_EMPS σDNO=5(EMPLOYEE)

RESULT1 ΠSSN(DEP5_EMPS)

RESULT2 ΠSUPERSSN(DEP5_EMPS)

RESULT RESULT1 U RESULT2

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CARTESIAN PRODUCT• Also called CROSS PRODUCT• R X S• R(A1, A2, . . ., An) X S(B1, B2, . . ., Bm) = Q(A1, A2, . . ., An, B1, B2,

. . ., Bm)• Attributes are in the order given above.• Q has one tuple for each combination of tuples in R and S.• n + m attributes• If R has x tuples and S has y tuples

» x * y tuples• Meaningless unless used with other operations

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• Example: Retrieve for each female employee a list of the names of her dependents.

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JOIN• Used to combine related tuples from two relations into single

tuples• Denoted by

R <join condition>S• Retrieve the name of the manager of each department.

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Example

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