Keys and Functional Dependency

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Keys and Functional Dependency
CS157A
Lecture 14

• Prof. Sin-Min Lee
• Department of Computer Science
• San Jose State University

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Data Normalization

• Primarily a tool to validate and improve a
  logical design so that it satisfies certain
  constraints that avoid unnecessary duplication of
  data.
• The process of decomposing relations with
  anomalies to produce smaller, well-structured
  relations.
• Primary Objective Reduce Redundancy,Reduce
  nulls,
• Improve modify activities
• insert,
• update,
• delete,
• but not read
• Price degraded query, display, reporting

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Functional Dependency and Keys

• Functional Dependency The value of one attribute
  (the determinant) determines the value of another
  attribute.
• Candidate Key Each non-key field is functionally
  dependent on every candidate key.

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Functional dependency

• a constraint between two attributes (columns) or
  two sets of columns
• A ? B if for every valid instance of A, that
  value of A uniquely determines the value of B
• or A ?B if there exists at most one value of B
  for every value of A

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Functional Dependencies
R
X Y Z

• FDs defined over two sets of attributes X, Y
  Ì R
• Notation X à Y reads as X determines Y
• If X à Y, then all tuples that agree on X must
  also agree on Y

1 2 3 2 4 5 1 2 4 1 2 7 2 4 8 3 7 9
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Functional Dependencies (example)
X Y Z
X Y Z
1 2 3 2 4 5 1 2 4 1 2 7 2 4 8 3 7 9
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functional dependency

• some examples
• SSN ? Name, Address, Birthdate
• VIN ? Make, Model, Color
• note the LHS is the determinant
• so functional dependency is the technical term
  for determines

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Candidate Keys

• an attribute (or set of attributes) that uniquely
  identifies a row
• primary key is a special candidate key
• values cannot be null
• e.g.
• ENROLL (Student_ID, Name, Address, )
• PK Student_ID
• candidate key Name, Address

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candidate key

• a candidate key must satisfy
• unique identification.
• implies that each nonkey attribute is
  functionally dependent on the key (for not(A ? B)
  to be true, A must occur more than once (with a
  different B), or A must map to more than one B in
  a given row)
• nonredundancy
• no attribute in the key can be deleted and still
  be unique
• minimal set of columns (Simsion)

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keys and dependencies
EMPLOYEE1 (Emp_ID, Name, Dept_Name, Salary)
determinant
Emp_ID Name Dept_Name Salary
functional dependency
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EMPLOYEE2 (Emp_ID, Course_Title, Name,
Dept_Name, Salary, Date_Completed)
Emp_ID Course_ Title Name Dept_ Name Salary Date_Comp.
not fully functionally dependant on the primary
key
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determinants candidate keys

• candidate key is always a determinant (one way to
  find a determinant)
• determinant may or may not be a candidate key
• ? candidate key is a determinant that uniquely
  identifies the remaining (nonkey) attributes
• determinant may be
• a candidate key
• part of a composite candidate key
• nonkey attribute

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Introduction

• Data integrity maintained by various constraints
  on data
• Functional dependencies are application
  constraints that help DB model real-world entity
• Join dependencies are a further constraint that
  help resolve some FD constraint limitations

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Normal Forms provide database designers with

• A formal framework for analyzing relation schemas
  based on their keys and on the functional
  dependencies among their attributes.
• A series of tests that can be carried out on
  individual relation schemas so that the
  relational database can be normalized to any
  degree.

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Keys

• superkeya superkey is a set of attributes S ?
  RA1,A2,.An with the property that no two
  tuples t1 and t2 in any relation state r of R
  will have t1S t2S.
• A key K is a superkey with the additional
  property that removal of any attribute from K
  will cause K not to be a superkey anymore.

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Keys

• The difference between a key and a superkey is
  that a key has to be minimal.
• Example
• SSN is a key for EMPLOYEE, whereas SSN,
  SSN,ENAME, SSN, ENAME, BDATE are all
  superkeys.

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Keys

• If a relation schema has more than one minimal
  key, each is called a candidate key.

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Keys

• one of the candidate keys is designated to be the
  primary key.
• Each relation schema must have a primary key.
• For example, SSN is the only candidate key for
  EMPLOYEE, so it is also the primary key.

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R(A B C D E)

• FD1. A -gt C
• FD2. BC -gtD
• FD3. E -gtAB
• result A
• By FD1. A -gt C A? result
• result A, C
• By FD2. BC -gt D BC? result
• ?result A, C
• By FD3. E -gtAB E? result
• ?result A, C
• ? A A, C

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• Similarly B B
• C C
• D D
• E E,A,B,C,D
• E is a candidate key
• Now, we see
• AB ABCD AC AC AD
  ACD
• BC BCD BD BD CD
  CD
• ABC ABCD ABD ABCD BCD
  BCD
• ACD ACD

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What is the largest normal form of this table?
R(A B C D E)
FD1. A -gtC FD2. BC -gtD FD3. E -gtAB Answer E
is the only candidate key of R The non-prime
attributes are A, B, C, D As FD!. A-gtC, we have
transitive dependency. Thus R(ABCD) is 2NF but
not 3NF
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What is Normalization?

• The purpose of normalization is to produce a
  stable set of relations that is a faithful model
  of the operations of the enterprise. By following
  the principles of normalization, we can achieve a
  design that is highly flexible, allowing the
  model to be extended when needed to account for
  new attributes, entity sets, and relationships.

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Normal Forms

• A relation is in specific normal form if it
  satisfies the set of requirements or constraints
  for that form. All of the normal forms are nested
  in that each satisfies the constraints of the
  previous one but is a "better" form because each
  eliminates flaws found in the previous

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Steps in Normalization
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1NF

• relation is in first normal form if it contains
  no multivalued attributes
• remove repeating groups to a new table as already
  demonstrated, carrying the PK as a FK

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First Normal Form ( 1NF )

• the domains of attributes must include only
  atomic(simple, indivisible) values and the value
  of any attribute in a tuple must be a single
  value from the domain of the attribute.

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First Normal Form ( 1NF )

• example
• Department
• DNAME DNUMBER DMGRSSN DLOCATIONS
• research 5 333445555
  Bellaire ,
• 
  Sugarland Houston
• Administration 4 987654321
  Stafford
• Headquarters 1 888665555
  Houston
• the domain of DLOCATIONS contains atomic values,
  but some tuples can have a set of these values.
  In this case,
• DNUMBER x-gtDLOCATIONS.
• The domain of DLOCATIONS contains sets of values
  and hence in non-atomic.

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Our Example in 1NF
PROJ_NUM
PROJ_NAME
EMP_NUM
EMP_NAME
JOB_CLASS
CHG_HOUR
HOURS

• Key (PROJ_NUM, EMP_NUM)
• Given PROJ_NUM
• PROJ_NAME is determined
• Given EMP_NUM
• EMP_NAME, JOB_CLASS, and CHG_HOUR are determined

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2NF

• a relation is in second normal form if it is in
  first normal form AND every nonkey attribute is
  fully functionally dependant on the primary key
• i.e. remove partial functional dependencies, so
  no nonkey attribute depends on just part of the
  key

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EMPLOYEE2 (Emp_ID, Course_Title, Name,
Dept_Name, Salary, Date_Completed)
Emp_ID Course_ Title Name Dept_ Name Salary Date_Comp.
not fully functionally dependant on the primary
key
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Second Normal Form ( 2NF )

• it is based on the concept of full functional
  dependency.
• A functional dependency X?Y is a full functional
  dependency , for any attribute A ? X, X - A
  ? Y.

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Second Normal Form

• A relation is in second normal form (2NF) if and
  only if it is in first normal form and all the
  nonkey attributes are fully functionally
  dependent on the key.

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Second Normal Form

• A table is in second normal form (2NF) if
• It is in 1NF
• It includes no partial dependencies. No
  attribute is dependent on only a portion of the
  primary key.

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2NF

• a relation is in 2NF if it is in 1NF and any one
  of these is true
• the PK consists of only 1 attribute
• all attributes are part of the PK (no nonkey
  attributes)
• every nonkey attribute is functionally dependant
  on the whole PK

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2NF (Example)
A B C D
R (A B C D) R (A B C D) R (A B C D) R (A B C D)
1 1 2 3
2 1 3 2
3 1 2 3
1 2 1 3
2 Candidate Keys
R with keyAB is NOT 2NF
R with keyAC is NOT 2NF
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Second Normal Form ( 2NF )
fd1
fd2
fd3

• SSN, PNUMBER?HOURS is a fully dependency
  (neither SSN?HOURS nor PNUMBER?HOURS holds).

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Second Normal Form ( 2NF )
EMP_PROJ
fd1
fd2
fd3
2NF NORMALIZATION
EP2
EP3
EP1
fd2
fd1
fd3

• The functional dependencies fd1,fd2,fd3 lead to
  the decomposition of EMP_PROJ into the three
  relation schemas EP1,EP2,EP3, each of which is in
  2NF.

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1NF ?2NF

• EMPLOYEE2 (Emp_ID, Course_Title, Name,
  Dept_Name, Salary, Date_Completed)
• ?
• EMPLOYEE1 (Emp_ID, Name, Dept_Name, Salary)
• and
• EMP_COURSE (Emp_ID, Course_Title, Date_Completed)
• EMPLOYEE1 satisfies condition1
• EMP_COURSE satisfies condition3

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3NF

• a relation is in third normal form if it is in
  2NF, AND no transitive dependencies exist
• transitive dependency is a functional dependency
  between nonkey attributes

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transitive dependency
transitive dependency
Cust_ID Name Salesperson Region
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transitive dependency

• same problems
• insertion anomaly (no salesman without a
  customer)
• deletion anomaly (if a salesman is assigned to
  only 1 customer, and the customer is deleted, we
  lose the salesman!)
• modification (update) anomaly (reassign
  salesperson to region)

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Cust_ID Name Salesperson Region
Salesperson Region
Cust_ID Name Salesperson
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Converting to 2NF

• To convert from 1NF to 2NF, list each key
  component and then the key itself.
• Each component will become the key in a new table.

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Our Example in 2NF
Table Name PROJECT
PROJ_NUM
PROJ_NAME
Table Name EMPLOYEE
CHG_HOUR
EMP_NUM
EMP_NAME
JOB_CLASS
Table Name ASSIGN
HOURS
PROJ_NUM
EMP_NUM
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Problems with 2NF

• Transitive Dependency
• An attribute that is dependent on a non-prime
  attribute exhibits transitive dependency.
• Still leads to data anomalies.

CHG_HOUR
EMP_NUM
EMP_NAME
JOB_CLASS
Our example contains the transitive
dependency JOB_CLASS -----gt CHG_HOUR
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Second Normal Form

• Second normal form
• Let R be a relation, and let F be the set of
  governing FDs. An attribute belongs to R is
  prime if a key of R contains A. In other words,
  A is prime in R if there exists KltR such that
  (1) K-gtR,
• (2) for all B belongs to K, (K-B)-gtR not
  belongs to F, and
• (3) A belongs to K

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Third Normal Form

• Third normal form
• Let R be a relation, a subset of the universal
  relation, in the context of a set of FDs F. R
  satisfies third normal form if for every
  nontrival X-gtA belong to F, either
• (1). X is superkey for R or
• (2). A is a prime attribute in R.

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Third Normal Form ( 3 NF )

• Third normal form is based on the concept of
  transitive dependency.
• A functional dependency X?Y in a relation schema
  R is a transitive dependency if there is a set of
  attributes Z that is not a subset of any key of
  R, and both X?Z and Z?Y hold.

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Third Normal Form ( 3 NF )
EMP_DEPT
ENAME SSN BDATE ADDRESS DNUMBER DNAME
DMGRSSN

• example
• the dependency SSN?DMGRSSN is transitive through
  DNUMBER in EMP_DEPT, because both the
  dependencies SSN?DNUMBER and DNUMBER?DMGRSSN hold
  and DNUMBER is not a subset of the key of
  EMP_DEPT.

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General Definitions of Second and Third Normal
Forms

• A relation schema R is in second normal form
  (2NF) if every nonprime attribute A in R is fully
  functionally dependent on every key of R.
• A relation schema R is in 3NF if, whenever a
  functional dependency X? A holds in R, either
• (a) X is a superkey of R.
• (b) A is a prime attribute of R.

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