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Functional Dependency


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What it is?

An attribute or set of attributes X is said tofunctionally determine another attribute

Y (written X Y) if and only if each X value is

associated with at most one Y value.

Customarily we call X determinant set andY a dependent set.

A functional dependency is a constraint between

two sets of attributes in a relation from a database.

So if we are given the value of X we can determine

the value of Y.


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The determination of functional dependencies

is an important part of designing databases inthe relational model, and in database normalization

and denormalization.

Why is Fanctional Dependency

Important?

The functional dependencies, along with the attribute

domains, are selected so as to generate constraintsthat would exclude as much data inappropriate to

the user domain from the system as possible.


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Inference Rules

Reflexivity:

If B is a subset of A then A functionally determines B

For example:

{name, location} {name}


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Inference Rules(cont.)

Augmentation:

If B is a subset of A and C functionally determines D

then A and C functionally determine B and D

For example:{name, location} and {birthdate, time} {name} and {age}

(as {name} is a subset of{name, location} and {birthdate, time}

functionally determines {age})


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Transitivity:

If A functionally determines B and B functionally

determines C then A functionally determines C

For example:{name, location} {initials}

(as {name, location} functionally determines

{name} and {name} functionally determines {initials})


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Pseudo transitivity:

If A functionally determines B and

B and C functionally determine D

then A and C functionally determine D


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Union:

If A functionally determines B and A functionally

determines C then A functionally determines B and C

For example:{name, location, birthdate, time} {initials, age}

(as {name, location, birthdate, time} {initials} and

{name, location, birthdate, time} {age})


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Inference Rules(cont.)Decomposition:

If A functionally determines B and Cthen A functionally determines B and

A functionally determines C

For example:{name, location, birthdate, time} {initials, age} implies that

{name, location, birthdate, time} {initials} and

{name, location, birthdate, time}{age}


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Trivial

Functional Dependencies

Some functional dependencies are said to be trivial

because they are satisfied by all relation.

For example:

A A

X Y if Y is a subset of X


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What is Superkey?

(review)

A superkey is defined in the relational model as a

set of attributes of a relation for which it holds that

in all instances of the relation there are no two

distinct tuples that have the same values for the

attributes in this set.

Equivalently a superkey can also be defined asthose sets of attributes of a relation upon which all

attributes of the relation are functionally dependent.


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Keys and Functional

Dependencies(cont.)

Functional dependencies allow us to express constraints

that we cannot express with superkeys.

Let's consider the schema of the example in the

textbook(p.265) Figure 7.2


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.

.

.

L-100..

.

loan_number amount

.

.

.

10000..

.

loan

.

.

.

23-652

15-20223-521.

.

.

customer_id loan_number

.

.

.

L-100

L-100L-100.

.

.

borrower

.

.

.

23-652

15-20223-521..

.

customer_id loan_number

.

.

.

L-100

L-100L-100.

.

.

amount

.

.

.

10000

1000010000.

.

.

bor_loan

Partial list of tuples in relations loan, borrower, and bor_loan


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Keys and Functional

Dependencies(contd.)

In previous figure, we consider the schema

bor_loan = (customer_id, loan_number, amount)

in which the functional dependency loan_number amountholdsbecause for each loan (identified by loan_number) there is a

unique amount. We denote the fact that the pair ofattributes(customer_id, loan_amount) forms a superkey for

bor_loan by writing:

customer_id, loan_number customer_id, loan_number, amount

or, equivalently,

customer_id, loan_number

bor_loan


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Keys and Functional

Dependencies(cont.)We shall use functional dependencies in two ways:

1. To test relations to see whether they are legal under

a given set of functional dependencies. If a relation ris a legal under a setFof functional dependencies,

we say that rsatisfies F.

2. To specify constraints on the set of legal relations.

We shall thus concern ourselves with only thoserelations that satisfy a given set of functional

dependencies. If we wish to constrain ourselves

to relations on schemaR that satisfy a setFof

functional dependencies, we say thatFholds onR.


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Database Normalization

Database normalization relates to reducing redundancy.

The key idea is to reduce the chance ofhaving multipledifferent versions ofthe same data, like degree, by storingall potentially duplicated data in different tables and linking

to them instead of using a single copy. Then updating thedegree in one place will instantly change all the placeswhere the degree is used.

updates can be quickly performed with little risk ofdata becoming inconsistent.

.


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Normal Forms

In the relational model, formal methods exist

to quantify "how to normalize a database ".

Such classifications are called normal forms,

and there are algorithms for converting a givendatabase between them.

Edgar F. Codd originally established 3 normal

forms:1NF, 2NF and 3NF.But 3NF is widely considered to be sufficient for

many practical applications. Most tables when

reaching 3NF are also in BCNF.

Normal Forms


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1NF(First Normal Form)

In the relational model, we formalize this idea that

attributes do not have any substructure. A domain

is atomic if elements of the domain are consideredto be indivisible units. We say that a relation schema

R is in first normal form if the domains of all

attributes ofR are atomic.


In the relational model, we formalize this idea that

attributes do not have any substructure. A domain

is atomic if elements of the domain are consideredto be indivisible units. We say that a relation schema

R is in first normal form if the domains of all

attributes ofR are atomic.


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To understand first normal form (1NF),consider these two examples of things you might know:

"What is your favourite colour?""What food will you not eat?"

Note difference between these two examples is that, you

can have only one favourite colour; but, little limitation on thenumber of foods you might not eat.

Data that has a single value such as "person's favourite

colour" is inherently in first normal form. Storing suchdata has not much changed since Codd wrote and needs

no further explanation here. Data that has multiple valuesmust be stored differently.


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2NF(Second Normal Form)

Second normal form (2NF) prescribes full functional

dependancy on the primary key. It most commonly

applies to tables that have composite primary keys,

where two or more attributes comprise the primary key.It requires that there are no non-trivial functional

dependencies of a non-key attribute on a part (subset)

of a candidate key.

A table is said to be in the 2NF if and only if it is in the

1NF and every non-key attribute is irreducibly

dependent on the primary key.


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2NF(Second Normal Form)

(cont.)How about this example?

Example:Machine_parts(part_number,supplier_name, price,supplier_address)

In this case, price is fully dependent on the primary

key(different suppliers may charge different price

on the same part).

However,supplier_address is partially dependent

because it only depends on thesupplier_name.

Therefore, this table is not in 2NF.


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3NF(Third Normal Form)

Third normal form (3NF) requires that there are no

non-trivial functional dependencies of non-key

attributes on something other than a superset of a

candidate key.

A table is in 3NF if it is in 2NF, and none of the non-

primary key attributes is a fact about any other non-

primary key attribute.

In summary, all non-key attributes are mutually

independent(i.e. there should not be transitive

dependencies).


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3NF(Third Normal Form)

(cont.)How about this example?

Example:Machine_parts(part_number,supplier_name,supplier_address)

In this case, supplier_address is dependent on

supplier_name. Therefore, there is a transitivedependency in the table.

It means that it is not in 3NF.


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BCNF

(Boyce-Codd Normal Form)Boyce-Codd normal form (orBCNF) requires that there are

no non-trivial functional dependencies of attributes onsomething else than a superset of a candidate key. At this stage,

all attributes are dependent on a key, a whole key and nothingbut a key (excluding trivial dependencies, like AA).

A table is said to be in the BCNF if and only if it is in the 3NF

and every non-trivial, left-irreducible functional dependencyhas a candidate key as its determinant.

In more informal terms, a table is in BCNF if it is in 3NF andthe only determinants are the candidate keys.


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Work Cited:

Silberschatz, Avi. HankKorth, and S. Sudarshan.

DATABASE SYSTEM CONCEPTS FIFTH

EDITION. New York:The McGraw-Hill

Companies, Inc., 205. 263-309.

Wikipedia:

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