copyright, harris corporation & ophir frieder, 19981 relational definitions

Copyright, Harris Corporation &

Ophir Frieder, 19981

Relational Definitions



Objectives

• To define basic relation terms including entity, attribute, domain, relation, relational scheme, and functional dependency.



Relational Terminology

• A domain is a set of scalar values, all of the same type.

Examples:

US city names - {San Diego, Miami, etc}

The names of US States - {Texas, New York, etc}

City populations - {1,2,3,...} (positive integers)



Relational Terminology, Cont.

• A relation is a subset of the Cartesian product of one or more domains.

Example:

{(San Diego,California,2655463), (Miami,Florida,3514403), (Syracuse,New York,745691)}




• A tuple is one member of a relation.

Example:

(Miami,Florida,3514403)




• It is helpful to view a relation as a table, where each row is one tuple and each column is one attribute.

Note that the relation may not actually correspond to one table in the final design - for the purposes of performance it may be combined with other tables, or broken into multiple tables.

• A relational scheme is the set of attributes associated with a relation.



Example - Relations

CITY STATE POPULATION

San Diego California 2655463

Miami Florida 3514403

Syracuse New York 745691



Functional Dependencies

• Depending on the meaning of the attributes some subsets of the Cartesian product are not valid relations.

CITY STATE POPULATION

San Diego California 2655463

Pittsburgh Florida 3514403

Syracuse New York 745691



Functional Dependencies, Cont.

• Let R be a relational scheme with attributes A1, A2,..., An .

• Let X and Y be subsets of {A1, A2,..., An}.

• We say that Y is functionally dependent on X in R, denoted X=>Y, if it is not possible for two distinct tuples in R to have the same value for the attributes in X, and different values for the attributes in Y. Alternatively, we say X functionally determines Y.

• Informally, if you know the left-hand-side, then you know the right-hand-side.




SS# => NAME

SS# => DATE-OF-BIRTH

SS# NAME DATE OF BIRTH

032446254 Mary Smith 11/4/68

045242453 Bob Jones 3/27/95

932415223 Sue Clark 8/3/76




• Note that the two functional dependencies:

SS# => NAME


• Could be represented by the single functional dependency:

SS# => NAME, DATE-OF-BIRTH

• Both representations have advantages, and will be used as appropriate.




• What are all of the possible functional dependencies for the relational scheme R=(SS#,NAME,DATE-OF-BIRTH)?

• Which of these hold “in the real world?”




• Functional dependencies are a result of what the attributes mean.

• A functional dependency is a statement about a relational scheme (i.e., all possible relations), and cannot be deduced from a particular relation.

• A functional dependency is said to be trivial if .

• Some functional dependencies can be enforced by a DataBase Management System (DBMS).

X Y X Y



Candidate KeyAn Informal Definition

• Let R be a relational scheme. A candidate key for R is a subset of the set of attributes of R, say K, such that

– Uniqueness property:

No two distinct tuples of R have the same value for K.

– Irreducibility property:

No proper subset of K has the uniqueness property.



Candidate Key, Cont.

• SS# is a candidate key for the following.

SS# => NAME


SS# NAME DATE OF BIRTH

032446254 Mary Smith 11/4/68

045242453 Bob Jones 3/27/95

932415223 Sue Clark 8/3/76



Candidate Key, Cont.

• Which of the following would be candidate keys?

SS#

NAME

DATE-OF-BIRTH

SS#, NAME

SS#, DATE-OF-BIRTH

NAME, DATE-OF-BIRTH



Candidate KeyA More Formal Definition

• Let R be a relational scheme with attributes A1, A2,..., An and functional dependencies F. In addition, let X be a subset of A1, A2,..., An. Then X is a candidate key for R if

– X => A1, A2,..., An is in F+, and

– for no proper subset Y of X is it the case that Y => A1, A2,..., An is in F+.



Closure Of A Set Of Dependencies

Question: What is F+?

Answer: F+ is the closure of F - the set of all dependencies derivable from F.

Equivalently, F+ is the set of all dependencies that follow from F by Armstrong’s

axioms.



Armstrong’s Axioms

Tangent!

Let R be a relational scheme with attributes U and functional dependencies F.

•Reflexivity - If Y is a subset of X, then X=>Y.

•Augmentation - If X=>Y, and Z is a subset of U, then XZ=>YZ.

•Transitivity - If X=>Y and Y=>Z, then X=>Z.



Keys - Additional Terminology

• Often times a relation will have more than one candidate key. In such cases one is sometimes designated as the primary key.

• Any superset of a candidate key is often times referred to as a superkey.

• Some authors will refer to any candidate key as simply a key, while others will use this to refer to a primary key.

• Throughout this course, the term key will be used synonymously with the phrase candidate key.

copyright, harris corporation & ophir frieder, 19981 relational definitions

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