Chapter 3

The Relational Model

Transparencies

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Chapter 3 - Objectives
◆ Terminology of relational model.
◆ How tables are used to represent data.
◆ Connection between mathematical relations
and relations in the relational model.
◆ Properties of database relations.
◆ How to identify CK, PK, and FKs.
◆ Meaning of entity integrity and referential
integrity.
◆ Purpose and advantages of views.

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Relational Model Terminology
◆ A relation is a table with columns and rows.
– Only applies to logical structure of the
database, not the physical structure.

◆ Attribute is a named column of a relation.

◆ Domain is the set of allowable values for one or

more attributes.

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Relational Model Terminology
◆ Tuple is a row of a relation.

◆ Degree is the number of attributes in a relation.

◆ Cardinality is the number of tuples in a relation.

◆ Relational Database is a collection of normalized

relations with distinct relation names.

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Instances of Branch and Staff Relations

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Examples of Attribute Domains

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Alternative Terminology for Relational Model

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Mathematical Definition of Relation
◆ Consider two sets, D1 & D2, where D1 = {2, 4} and D2
= {1, 3, 5}.
◆ Cartesian product, D1 × D2, is set of all ordered pairs,
where first element is member of D1 and second
element is member of D2.

D1 × D2 = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}

◆ Alternative way is to find all combinations of elements

with first from D1 and second from D2.
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Mathematical Definition of Relation
◆ Any subset of Cartesian product is a relation; e.g.
R = {(2, 1), (4, 1)}
◆ May specify which pairs are in relation using
some condition for selection; e.g.
– second element is 1:
R = {(x, y) | x ∈ D1, y ∈ D2, and y = 1}
– first element is always twice the second:
S = {(x, y) | x ∈ D1, y ∈ D2, and x = 2y}

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Mathematical Definition of Relation
◆ Consider three sets D1, D2, D3 with Cartesian
Product D1 × D2 × D3; e.g.

D1 = {1, 3} D2 = {2, 4} D3 = {5, 6}

D1 × D2 × D3 = {(1,2,5), (1,2,6), (1,4,5), (1,4,6),
(3,2,5), (3,2,6), (3,4,5), (3,4,6)}

◆ Any subset of these ordered triples is a relation.

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Mathematical Definition of Relation
◆ Cartesian product of n sets (D1, D2, . . ., Dn) is:

D1 × D2 × . . . × Dn = {(d1, d2, . . . , dn) | d1 ∈ D1, d2

∈ D2, . . . , dn∈ Dn}
usually written as:
n
X Di
i=1

◆ Any set of n-tuples from this Cartesian product is

a relation on the n sets.

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Database Relations
◆ Relation schema
– Named relation defined by a set of attribute
and domain name pairs.

◆ Relational database schema

– Set of relation schemas, each with a distinct
name.

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Properties of Relations
◆ Relation name is distinct from all other relation
names in relational schema.

◆ Each cell of relation contains exactly one atomic

(single) value.

◆ Each attribute has a distinct name.

◆ Values of an attribute are all from the same

domain.

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Properties of Relations
◆ Each tuple is distinct; there are no duplicate
tuples.

◆ Order of attributes has no significance.

◆ Order of tuples has no significance, theoretically.

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Relational Keys
◆ Superkey
– An attribute, or set of attributes, that uniquely
identifies a tuple within a relation.

◆ Candidate Key
– Superkey (K) such that no proper subset is a superkey
within the relation.
– In each tuple of R, values of K uniquely identify that
tuple (uniqueness).
– No proper subset of K has the uniqueness property
(irreducibility).

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Relational Keys
◆ Primary Key
– Candidate key selected to identify tuples uniquely
within relation.

◆ Alternate Keys
– Candidate keys that are not selected to be primary
key.

◆ Foreign Key
– Attribute, or set of attributes, within one relation
that matches candidate key of some (possibly same)
relation.
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Integrity Constraints
◆ Null
– Represents value for an attribute that is
currently unknown or not applicable for tuple.
– Deals with incomplete or exceptional data.
– Represents the absence of a value and is not the
same as zero or spaces, which are values.

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Integrity Constraints
◆ Entity Integrity
– In a base relation, no attribute of a primary
key can be null.

◆ Referential Integrity
– If foreign key exists in a relation, either
foreign key value must match a candidate
key value of some tuple in its home relation
or foreign key value must be wholly null.

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Integrity Constraints
◆ General Constraints
– Additional rules specified by users or
database administrators that define or
constrain some aspect of the enterprise.

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Views
◆ Base Relation
– Named relation corresponding to an entity
in conceptual schema, whose tuples are
physically stored in database.

◆ View
– Dynamic result of one or more relational
operations operating on base relations to
produce another relation.

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Views
◆ A virtual relation that does not necessarily actually
exist in the database but is produced upon request,
at time of request.

◆ Contents of a view are defined as a query on one or

more base relations.

◆ Views are dynamic, meaning that changes made to

base relations that affect view attributes are
immediately reflected in the view.

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Purpose of Views
◆ Provides powerful and flexible security
mechanism by hiding parts of database from
certain users.

◆ Permits users to access data in a customized way,

so that same data can be seen by different users
in different ways, at same time.

◆ Can simplify complex operations on base

relations.

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Updating Views
◆ All updates to a base relation should be
immediately reflected in all views that
reference that base relation.

◆ If view is updated, underlying base relation

should reflect change.

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Updating Views
◆ There are restrictions on types of modifications that
can be made through views:
– Updates are allowed if query involves a single base
relation and contains a candidate key of base
relation.
– Updates are not allowed involving multiple base
relations.
– Updates are not allowed involving aggregation or
grouping operations.

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Updating Views
◆ Classes of views are defined as:
– theoretically not updateable;
– theoretically updateable;
– partially updateable.

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