CS 377

Database Systems

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Relational Data Model

Li Xiong

Department of Mathematics and Computer Science

Emory University

Outline

� Relational Model Concepts

� Relational Model Constraints

� Relational Database and operations

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Relational Model

� First formal database model

� Introduced by Codd in "A Relational Model for

Large Shared Data Banks," Communications of

the ACM, June 1970.

� First commercial implementations available in

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� First commercial implementations available in

early 1980s

� Based on the concept of a mathematical relation

and has theoretical basis in set theory and first-

order predicate logic.

� Other models: hierarchical model, network model

INFORMAL DEFINITIONS

� RELATION: A table of values

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

� A relation may alternately be thought of as a set of columns.

� Each row represents a fact that corresponds to a real-world entity or relationship.

� Table name and column names help interpret the meaning of the values

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FORMAL DEFINITIONS – Relation Schema

� The table is called a relation, a row is a tuple, a column header is an attribute

� Relation Schema R (A1, A2, .....An)

� Made up of a relation name R and a set of attributes A1, A2, .....An

� Degree (or arity) of a relation is the number of attributes n of its relational schema

� E.g. STUDENT (Name, SSN, HomePhone, Address, OfficePhone, Age, GPA)

� Each attribute Ai has a domain dom(Ai) that defines the possible values of the

attribute by a data-type or a format

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attribute by a data-type or a format

� E.g. The domain of SSN is the set of 9 digit numbers defined as: ddd-dd-dddd

where each d is a decimal digit.

FORMAL DEFINITIONS – Relation

� A relation (or relation state) r of the relation schema R (A1, A2,

.....An), r(R), is a set of tuples r = {t1, t2, ..., tm}

� A tuple t is an ordered set of n values t =<v1, v2, ..., vn>, where

each value vi, 1 ≤ i ≤ n, is an element of dom(Ai) or a special

NULL value

� E.g. <“Benjamin Bayer", 305-61-2435, 373-1616, “2918 Bluebonnet Lane”, null,

19, 3.21> is a tuple belonging to the STUDENT relation.

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19, 3.21> is a tuple belonging to the STUDENT relation.

Mathematical Definitions� A relation r(R) is a mathematical relation of degree n on the domains

dom(A1), dom(A2), ..., dom(An), which is a subset of the Cartesian

product of the domains that define R:

r(R) ⊆ (dom(A1) × dom(A2) × ... × dom(An))

� The Cartesian product is the direct product of the sets of values of all domains: dom (A1) × dom (A2) × ... × dom(An)

� The total number of tuples in the Cartesian product is:

|dom (A )| × |dom (A )| × ... × |dom(A )|

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|dom (A1)| × |dom (A2)| × ... × |dom(An)|

� Current relation state reflects only the valid tuples that represent a particular state

CHARACTERISTICS OF RELATIONS

� Ordering of tuples in a relation r(R) � A relation is a set of tuples which are not ordered

� Ordering of attributes� The attributes in R(A1, A2, ..., An) and the values in t=<v1, v2, ..., vn> are an ordered

list in our definition

� Alternative definition: tuple considered as a set of (<attribute>, <value>) pairs,

where each pair gives the value of the mapping from an attribute Ai to a value vi from dom(Ai)

� Values in a tuple

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Values in a tuple� All values are considered atomic (flat relational model with first normal form

assumption) – what about multi-valued attributes and composite attributes?

� A special null value is used to represent values that are unknown or inapplicable to certain tuples.

DEFINITION SUMMARY

Informal Terms Formal Terms

Table Relation

Column Attribute

Row Tuple

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Row Tuple

Values in a column Domain

Table Definition Relation Schema

Relational Model Notation� Relation schema R of degree n: R(A1, A2, ..., An)

� Relation names: Q, R, S

� Relations: q, r, s

� Tuples: t, u, v

� tuple t in a relation r(R): t = <v , v , ..., v >, v is the

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� tuple t in a relation r(R): t = <v1, v2, ..., vn>, vi is the

value corresponding to attribute Ai

� Component values of tuples:

� t[Ai] and t.Ai refer to the value vi in t for attribute Ai

� t[Au, Aw, ..., Az] and t.(Au, Aw, ..., Az) refer to the subtuple

of values <vu, vw, ..., vz> from t corresponding to the

attributes specified in the list

Outline

� Relational Model Concepts

� Relational Model Constraints

� Relational Database and operations

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Relational Model Constraints� Constraints

� Restrictions on the actual values in a database state

� Inherent model-based constraints or implicit constraints

� Inherent in the data model

� E.g. no duplicate tuples

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� Schema-based constraints or explicit constraints

� Can be directly expressed in schemas of the data model

� Application-based or semantic constraints or business

rules

� Cannot be directly expressed in schemas, expressed and enforced by

application program

� E.g. the max. no. of hours per employee for all projects he or she

works on is 56 hrs per week

Schema-based constraints

� Domain constraints

� Key constraints

� Entity integrity constraints

� Referential integrity constraints

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� Referential integrity constraints

Domain Constraints

� The value of each attribute A must be an atomic value

from the domain dom(A)

� Typical data types associated with domains:

� Numeric data types for integers and real numbers

Characters

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� Characters

� Booleans

� Fixed-length strings

� Variable-length strings

� Date, time, timestamp

� Money

� Other special data types

Key Constraints

� No two tuples can have the same combination of

values for all their attributes.

� Superkey

� No two distinct tuples in any state r of R can have the

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� No two distinct tuples in any state r of R can have the

same value for SK

� Key

� Superkey of R

� Removing any attribute A from K leaves a set of

attributes K that is not a superkey of R any more

Key Constraints and Constraints

on NULL Values (cont’d.)� Key satisfies two properties:

� Two distinct tuples in any state of relation cannot have

identical values for (all) attributes in key

� Minimal superkey

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• Cannot remove any attributes and still have uniqueness

constraint in above condition hold

Key Constraints and Constraints

on NULL Values (cont’d.)� Candidate key

� Relation schema may have more than one key

� Primary key of the relation

� Designated among candidate keys

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� Designated among candidate keys

� Underline attribute

� Other candidate keys are designated as unique keys

Key Constraints and Constraints on

NULL Values (cont’d.)

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Key Constraints

� Superkey of R: A set of attributes SK of R such that no two tuplesin any valid relation instance r(R) will have the same value for SK.

� For any distinct tuples t1 and t2 in r(R), t1[SK] ≠≠≠≠ t2[SK].

� {Licence_number}, {License_number, Make}, {Engine_serial_number, Make}

� 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.

� Key1 = {License_number}, Key2 = {Engine_serieal_number}

� Is {Engine_serial_number, Make} a key?

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� Is {Engine_serial_number, Make} a key?

� If a relation has several keys, each is called a candidate key, and one is chosen arbitrarily to be the primary key. The primary key attributes are underlined.

Entity Integrity

� Entity Integrity: The primary key attributes PK of each relation schema R cannot have null values in any tuple of r(R).

t[PK] ≠≠≠≠ null for any tuple t in r(R)

� Primary key values are used to identify the individual tuples.

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

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null values, even though they are not members of the primary key.

Referential Integrity

� Referential integrity: a tuple in one relation that refers to another relation must refer to an existing tuple in that relation.

� Formally

A set of attributes FK in relation schema R1 is a foreign key of R1 that references relation R2 if:� FK have the same domains as the primary key attributes PK of R2

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� The value of FK in the current state of R1 can be either: � (1) a value of PK in the current state of R2: t1[FK] = t2[PK].

� (2) a null

� R1 is the referencing relation and R2 is the referenced relation.

� 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.

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Outline

� Relational Model Concepts

� Relational Model Constraints

� Relational Database and operations

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Relational Databases and Relational

Database Schemas� Relational database schema S

� Set of relation schemas S = {R1, R2, ..., Rm}

� Set of integrity constraints IC

� Relational database state

� Set of relation states DB = {r , r , ..., r }

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� Set of relation states DB = {r1, r2, ..., rm}

� Each ri is a state of Ri such that the ri relation states satisfy integrity

constraints specified in IC

� Invalid state

� Does not obey all the integrity constraints

� Valid state

� Satisfies all the constraints in the defined set of integrity constraints

IC

Operations in a Relational Database� Basic operations that change the states of relations in

the database:

� Insert

� Delete

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� Update (or Modify)

The Insert Operation� Provides a list of attribute values for a new tuple t

to be inserted into a relation R

� Can violate any of the four types of constraints

� Default option is to reject the insertion

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The Delete Operation� Can violate only referential integrity

� If tuple being deleted is referenced by foreign keys from other tuples, e.g.

delete a tuple from department

� Restrict Reject the deletion

� Cascade Propagate the deletion by deleting tuples that reference the deleted

tuple

� Set null or set default Modify the referencing attribute values that cause the

violation

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violation

The Update Operation

� Necessary to specify a condition on attributes of

relation

� Select the tuple (or tuples) to be modified

� If attributes to be updated not part of a primary key

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� If attributes to be updated not part of a primary key

nor of a foreign key

� Usually causes no problems

� Updating a primary/foreign key

� Similar issues as with Insert/Delete

In-Class Exercise

Consider the following relations for a database that keeps

track of student enrollment in courses and the books adopted

for each course:

STUDENT(SSN, Name, Major, Bdate)

COURSE(Course#, Cname, Dept)

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COURSE(Course#, Cname, Dept)

ENROLL(SSN, Course#, Quarter, Grade)

BOOK_ADOPTION(Course#, Quarter, Book_ISBN)

TEXT(Book_ISBN, Book_Title, Publisher, Author)

Draw a relational schema diagram specifying the foreign

keys for this schema.