Chapter 8 Functional Dependencies and Normalization for Relational Databases

Chapter 8Functional Dependencies and Normalization for Relational Databases

Chapter 10-2

Chapter Outline1 Informal Design Guidelines for Relational Databases

1.1Semantics of the Relation Attributes1.2 Redundant Information in Tuples and Update Anomalies1.3 Null Values in Tuples1.4 Spurious Tuples

2 Functional Dependencies (FDs)2.1 Definition of FD2.2 Inference Rules for FDs2.3 Equivalence of Sets of FDs2.4 Minimal Sets of FDs

Chapter 10-3

Chapter Outline

3 Normal Forms Based on Primary Keys3.1 Normalization of Relations

3.2 Practical Use of Normal Forms

3.3 Definitions of Keys and Attributes Participating in Keys

3.4 First Normal Form

3.5 Second Normal Form

3.6 Third Normal Form

4 General Normal Form Definitions (For Multiple Keys)

5 BCNF (Boyce-Codd Normal Form)

Chapter 10-4

1 Informal Design Guidelines for Relational Databases (1)

What is relational database design?

The grouping of attributes to form "good" relation schemas

 Two levels of relation schemas The logical "user view" level The storage "base relation" level

 Design is concerned mainly with base relations  What are the criteria for "good" base relations? 

Chapter 10-5

Informal Design Guidelines for Relational Databases (2)

We first discuss informal guidelines for good relational design

Then we discuss formal concepts of functional dependencies and normal forms- 1NF (First Normal Form)- 2NF (Second Normal Form)- 3NF (Third Normal Form)- BCNF (Boyce-Codd Normal Form)

Additional types of dependencies, further normal forms, relational design algorithms by synthesis are discussed in Chapter 11

Chapter 10-6

Semantics of the Relation Attributes

GUIDELINE 1: Informally, each tuple in a relation should represent one entity or relationship instance. (Applies to individual relations and their attributes).

Attributes of different entities (EMPLOYEEs, DEPARTMENTs, PROJECTs) should not be mixed in the same relation

Only foreign keys should be used to refer to other entities  Entity and relationship attributes should be kept apart as much as

possible.

Chapter 10-7

A simplified COMPANY relational database schema

Chapter 10-8

Redundant Information in Tuples and Update

Anomalies Mixing attributes of multiple entities may cause problems Information is stored redundantly wasting storage Problems with update anomalies

Insertion anomalies Deletion anomalies Modification anomalies

University database schemas example

sno cname grade

sdept dname

A department has some students. A student only belongs to a department. A department only has one dean. A student can choose many courses and a course can be chosen by many students. Every student has his grade of each course.

Assuming that:

The relationships are ：

student （ sno,sdept,dname,cname,grade ）

Example database instances

sno sdept dname cname grade

080640101 cs Mr Wu DB 87

080640102 cs Mr Wu DB 89

080640101 cs Mr Wu Data strucure

88

080640104 cs Mr Wu DB 85

080640105 cs Mr Wu DB 84

Bad influences

Relation schemas ： student （ sno,sdept,dname,cname,grade ） has some problems:

1. redundancy sdept,dname,cname repetition 2. update anomalies 3. insert anomalies 4. deletion anomalies why ？

Because we have some data dependencies.

Chapter 10-12

Guideline to Redundant Information in Tuples and Update Anomalies

GUIDELINE 2: Design a schema that does not suffer from the insertion, deletion and update anomalies. If there are any present, then note them so that applications can be made to take them into account

Chapter 10-13

Null Values in Tuples

GUIDELINE 3: Relations should be designed such that their tuples will have as few NULL values as possible

 Attributes that are NULL frequently could be placed in separate relations (with the primary key)

 Reasons for nulls: attribute not applicable or invalid attribute value unknown (may exist) value known to exist, but unavailable

Chapter 10-14

Spurious Tuples

Bad designs for a relational database may result in erroneous results for certain JOIN operations

The "lossless join" property is used to guarantee meaningful results for join operations

GUIDELINE 4: The relations should be designed to satisfy the lossless join condition. No spurious tuples should be generated by doing a natural-join of any relations.

Chapter 10-15

Spurious Tuples

 There are two important properties of decompositions:

(a) non-additive or losslessness of the corresponding join

(b) preservation of the functional dependencies.

Note that property (a) is extremely important and cannot be sacrificed. Property (b) is less stringent and may be sacrificed.

Chapter 10-16

Functional Dependencies (1) Functional dependencies (FDs) are used to specify formal

measures of the "goodness" of relational designs FDs and keys are used to define normal forms for relations FDs are constraints that are derived from the meaning and

interrelationships of the data attributes. That is we determine FDs from understanding the data items and rules of the enterprise and not by mere observation of the rows in the tables.

X->Y : A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y.

Chapter 10-17

Functional Dependencies (2)

X -> Y holds if whenever two tuples have the same value for X, they must have the same value for Y

For any two tuples t1 and t2 in any relation instance r(R): If t1[X]=t2[X], then t1[Y]=t2[Y]

X -> Y in R specifies a constraint on all relation instances r(R) Written as X -> Y; can be displayed graphically on a relation schema as

in Figures. ( denoted by the arrow: ).

FDs are derived from the real-world constraints on the attributes

Chapter 10-18

Examples of FD constraints (1)

social security number determines employee name

SSN -> ENAME project number determines project name and location

PNUMBER -> {PNAME, PLOCATION} employee ssn and project number determines the hours per

week that the employee works on the project

{SSN, PNUMBER} -> HOURS

Chapter 10-19

Examples of FD constraints (2)

An FD is a property of the attributes in the schema R The constraint must hold on every relation instance r(R) If K is a key of R, then K functionally determines all

attributes in R (since we never have two distinct tuples with t1[K]=t2[K])

A FD XY is trivial if Y is a proper subset of X; otherwise it is nontrivial.

Chapter 10-20

Inference Rules for FDs (1) Given a set of FDs F, we can infer additional FDs that hold

whenever the FDs in F hold

 Armstrong's inference rules:IR1. (Reflexive) If Y subset-of X, then X -> Y

IR2. (Augmentation) If X -> Y, then XZ -> YZ

(Notation: XZ stands for X U Z or {X, Z})

IR3. (Transitive) If X -> Y and Y -> Z, then X -> Z

IR1, IR2, IR3 form a sound and complete set of inference rules

Chapter 10-21

Inference Rules for FDs (2)

Some additional inference rules that are useful:(Decomposition) If X -> YZ, then X -> Y and X -> Z(Union) If X -> Y and X -> Z, then X -> YZ(Psuedotransitivity) If X -> Y and WY -> Z, then WX -> Z

The last three inference rules, as well as any other inference rules, can be deduced from IR1, IR2, and IR3 (completeness property)

See page 309 for the proof or use an exhaustive tree to prove it by yourself (forward or backtrack)

Chapter 10-22

Inference Rules for FDs (3)

Closure of a set F of FDs is the set F+ of all FDs that can be inferred from F

Closure of a set of attributes X with respect to F is the set X + of all attributes that are functionally determined by X

X + can be calculated by repeatedly applying IR1, IR2, IR3 using the FDs in F

Chapter 10-23

Equivalence of Sets of FDs

Two sets of FDs F and G are equivalent if:- every FD in F can be inferred from G, and- every FD in G can be inferred from F

Hence, F and G are equivalent if F + =G +

Definition: F covers G if every FD in G can be inferred from F (i.e., if G + subset-of F +)

F and G are equivalent if F covers G and G covers F There is an algorithm for checking equivalence of sets of FDs

See page 310 for the algorithm and the example.

Chapter 10-24

Minimal Sets of FDs (1) It is possible to have sets of FDs that might look completely diff

erent but nonetheless be logically equivalent. Since there might be many sets of FDs equivalent to any given set, we question whether there is a set of FDs that can be viewed as “canonical”. It turns out that defining a unique canonical set is not an easy task, but the notion of minimal cover comes close.

Chapter 10-25

Minimal Sets of FDs (1) Minimal Cover Let F be a set of FDs. A minimal cover of F is a set of FDs, G, that has the fo

llowing properties: G is equivalent to F (but, possibly, different from F) All FDs in G have the form X A, where A is a single attribute. That is

every dependency in G has a single attribute for its RHS. It is not possible to make G “smaller” (and still satisfy the first two propert

ies)by either of the following: Deleting an FD Deleting an attribute from an FD

NOTE: We cannot replace any dependency X -> A in F with a dependency Y -> A, where Y proper-subset-of X ( Y subset-of X) and still have a set of dependencies that is equivalent to G.

Chapter 10-26

Minimal Sets of FDs (2)

Every set of FDs has an equivalent minimal set There can be several equivalent minimal sets There is no simple algorithm for computing a minimal set of

FDs that is equivalent to a set F of FDs To synthesize a set of relations, we assume that we start with a

set of dependencies that is a minimal set

Minimal Cover Algorithm

Input: A set of FDs F

1. Output: G , a minimal cover of F

2. Step 1: G=F, where all FDs are converted to use singleton-attributes on the RHS

3. Remove all redundant attributes from the LHS of FDs in G

4. Remove all redundant FDs from G Return G

Chapter 10-27

Minimal Sets of FDs (3)Consider the attribute set ABCDEFGH and the following set, F, of FDs: ABH C, A D, C E, BGH F,

F AD, E F, BH E

Since not all right-hand sides are single attributes, we can use the decomposition rule to obtain an FD set that satisfies

the first two properties of minimal covers.

ABH C, A D, C E, BGH F, F AD {F A, F D }, E F, BH E

We can see that BGH F is entailed by BH E and E F, and that F D is entailed by F A and A D. Thus,

we are left with. ABH C, A D, C E, F A, E F, BH E

It is easy to check by computing attribute closures that none of these FDs is redundant; that is, one cannot simply throw

out an FD from this set without sacrificing equivalence to the original set F.

It is possible to delete the attribute A from ABH C. This is because of the ff FDs: BH E, E F, , F A .

Hence we can infer that BH A.

We now have the ff FDs: BH C, A D, C E, F A, E F, BH E

Compute the closure of ABH and BH and compare

Again is redundant since BH E can be inferred from BH C, C E hence it can be removed. Therefore

the minimal cover is BH C, A D, C E, F A, E F

Chapter 10-28

Exercises

R{A,B,C},F:{A->BC,B->C,A->B,AB->C} FMIN ? (the first step: single right attribute) (the second step: delete the redundant FDs) FMIN ? ={ B->C,A->B}

Chapter 10-30

3 Normal Forms Based on Primary Keys 3.1 Normalization of Relations

3.2 Practical Use of Normal Forms

3.3 Definitions of Keys and Attributes Participating in Keys

3.4 First Normal Form

3.5 Second Normal Form

3.6 Third Normal Form

3.7 BCNF

Chapter 10-31

Normalization of Relations (1)

Normalization: The process of decomposing unsatisfactory "bad" relations by breaking up their attributes into smaller relations

Normal form: Condition using keys and FDs of a relation to certify whether a relation schema is in a particular normal form

Chapter 10-32

Normalization of Relations (2)

2NF, 3NF, BCNF based on keys and FDs of a relation schema 4NF based on keys, multi-valued dependencies : MVDs; 5NF

based on keys, join dependencies : JDs (Chapter 11) Additional properties may be needed to ensure a good

relational design (lossless join, dependency preservation; Chapter 11)

Chapter 10-33

Practical Use of Normal Forms

Normalization is carried out in practice so that the resulting designs are of high quality and meet the desirable properties

The practical utility of these normal forms becomes questionable when the constraints on which they are based are hard to understand or to detect

The database designers need not normalize to the highest possible normal form. (usually up to 3NF, BCNF or 4NF)

Denormalization: the process of storing the join of higher normal form relations as a base relation—which is in a lower normal form

Chapter 10-34

Definitions of Keys and Attributes Participating in Keys (1)

A superkey of a relation schema R = {A1, A2, ...., An} is a set of attributes S subset-of R with the property that no two tuples t1 and t2 in any legal relation state r of R will have t1[S] = t2[S]

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 any more. A key is a minimal superkey

Chapter 10-35

Definitions of Keys and Attributes Participating in Keys (2)

If a relation schema has more than one key, each is called a candidate key. One of the candidate keys is arbitrarily designated to be the primary key, and the others are called secondary keys.

A Prime attribute must be a member of some candidate key A Nonprime attribute is not a prime attribute—that is, it is

not a member of any candidate key.

Chapter 10-36

First Normal Form

Disallows composite attributes, multivalued attributes, and nested relations; attributes whose values for an individual tuple are non-atomic

Considered to be part of the definition of relation

Chapter 10-37

Figure Normalization into 1NF

Chapter 10-38

Second Normal Form (1) Uses the concepts of FDs, primary keyDefinitions: Prime attribute - attribute that is member of the primary

key K Full functional dependency - a FD Y -> Z where removal

of any attribute from Y means the FD does not hold any moreExamples: - {SSN, PNUMBER} -> HOURS is a full FD since neither SSN -> HOURS nor PNUMBER -> HOURS hold - {SSN, PNUMBER} -> ENAME is not a full FD (it is called a partial dependency ) since SSN -> ENAME also holds

Chapter 10-39

Second Normal Form (2)

A relation schema R is in second normal form (2NF) if every non-prime attribute A in R is fully functionally dependent on the primary key

R can be decomposed into 2NF relations via the process of 2NF normalization

Figure Normalizing into 2NF and 3NF

Third Normal Form (1)

Definition: Transitive functional dependency - a FD X -> Z that can be

derived from two FDs X -> Y and Y -> Z

Examples:

- SSN -> DMGRSSN is a transitive FD since

SSN -> DNUMBER and DNUMBER -> DMGRSSN hold

- SSN -> ENAME is non-transitive since there is no set of attributes X where SSN -> X and X -> ENAME

Figure Normalizing into 2NF and 3NF

Third Normal Form (2)

A relation schema R is in third normal form (3NF) if it is in 2NF and no non-prime attribute A in R is transitively dependent on the primary key

R can be decomposed into 3NF relations via the process of 3NF normalization

NOTE:In X -> Y and Y -> Z, with X as the primary key, we consider this a problem only if Y is not a candidate key. When Y is a candidate key, there is no problem with the transitive dependency .E.g., Consider EMP (SSN, Emp#, Salary ). Here, SSN -> Emp# -> Salary and Emp# is a candidate key.

Chapter 10-44

4 General Normal Form Definitions (For Multiple Keys) (1)

The above definitions consider the primary key only The following more general definitions take into account

relations with multiple candidate keys A relation schema R is in second normal form (2NF) if

every non-prime attribute A in R is fully functionally dependent on every key of R

Chapter 10-45

General Normal Form Definitions (2)

Definition: Superkey of relation schema R - a set of attributes S of R that

contains a key of R A relation schema R is in third normal form (3NF) if

whenever a FD X -> A holds in R, then either: (a) X is a superkey of R, or (b) A is a prime attribute of R

NOTE: Boyce-Codd normal form disallows condition (b) above

5 BCNF (Boyce-Codd Normal Form)

A relation schema R is in Boyce-Codd Normal Form (BCNF) if whenever an FD X -> A holds in R, then X is a superkey of R

Each normal form is strictly stronger than the previous one Every 2NF relation is in 1NF Every 3NF relation is in 2NF Every BCNF relation is in 3NF

There exist relations that are in 3NF but not in BCNF The goal is to have each relation in BCNF (or 3NF)

Figure Boyce-Codd normal form

Figure a relation TEACH that is in 3NF but not in BCNF

EXAMPLE

sno

cno

sdept

sloc

grade

The 1NF relation schemas may be not good relation schemas.

SLC （ sno,sdept,sloc,cno,grade ） primary key （ sno,cno ）。

Problems: Redundancy ； insertion anomalies ； update anomalies ； deletion anomalies

sno sdept sloc cno grade

080640101 cs 11 building 01 89

080640102 cs 11 building 01 90

080640103 cs 11 building 02 86

080640103 cs 11 building 01 88

080640101 cs 11 building 02 87

Decomposition of 1NF

SLC （ sno,sdept,sloc,cno,grade ） changed to ： SL(sno,sdept,sloc) SC(sno,cno,grade) It is removal of the partial FD between sdept to sn

o

sno gradesno

cno

sdept

sloc

SLC （ sno,sdept,sloc,cno,grade ） changed to ： SL(sno,sdept,sloc)

SC(sno,cno,grade) Now the relation schemas is in 2NF.

But SL(sno,sdept,sloc) still has deletion anomalies,update anomalies,insertion anomalies.

sno sdept sloc

080640101 cs 11 building

080640102 cs 11 building

080640103 cs 11 building

080640104 cs 11 building

080640105 cs 11 building

Decomposition of 2NF

SL （ sno,sdept,sloc ） remove the transitive FD between sloc and sno ：

SD （ sno,sdept ） DL （ sdept,sloc ） Resolve some problems ： redundancy , insertion anomalie

s, deletion anomalies, update anomalies.

sno

sdept

sloc

sno sdept

slocsdept

Application

(1) R(WXYZ) primary key is WX.we still have X->Z Decompose to 2NF R1(XZ) R2(WXY) Repeate this process until delete all the partia

l functional dependancy.

Application

R(WXY), primary key is W. we still have X->Y. Decompose this to 3nf. R1(XY) R2(WX)

1. Properties of Relational Decompositions (1)

Relation Decomposition and Insufficiency of Normal Forms:

Universal Relation Schema: a relation schema R={A1, A2, …, An} that includes all the attributes of the database.

Universal relation assumption: every attribute name is unique. Decomposition: The process of decomposing the universal relation sch

ema R into a set of relation schemas D = {R1,R2, …, Rm} that will become the relational database schema by using the functional dependencies.

Properties of Relational Decompositions (2)

Relation Decomposition and Insufficiency of Normal Forms (cont.):

Attribute preservation condition: Each attribute in R will appear in at least one relation schema Ri in the decomposition so that no attributes are “lost”.

Another goal of decomposition is to have each individual relation Ri in the decomposition D be in BCNF or 3NF.

Additional properties of decomposition are needed to prevent from generating spurious tuples

Properties of Relational Decompositions (3)

Dependency Preservation Property of a Decomposition : Definition: Given a set of dependencies F on R, the projection of F on Ri, d

enoted by pRi(F) where Ri is a subset of R, is the set of dependencies X Y in F+ such that the attributes in X υ Y are all contained in Ri. Hence, the projection of F on each relation schema Ri in the decomposition D is the set of functional dependencies in F+, the closure of F, such that all their left- and right-hand-side attributes are in Ri.

Properties of Relational Decompositions (4)

Dependency Preservation Property of a Decomposition (cont.): Dependency Preservation Property: a decomposition D = {R1,

R2, ..., Rm} of R is dependency-preserving with respect to F if the union of the projections of F on each Ri in D is equivalent to F; that is, ((R1(F)) υ . . . υ (Rm(F)))+ = F+ (See examples in Fig 10.12a and Fig 10.11)

Claim 1: It is always possible to find a dependency-preserving decomposition D with respect to F such that each relation Ri in D is in 3nf.

Properties of Relational Decompositions (5)

Lossless (Non-additive) Join Property of a Decomposition: Definition: Lossless join property: a decomposition D = {R1, R2, ..., Rm} of R has the

lossless (nonadditive) join property with respect to the set of dependencies F on R if, for every relation state r of R that satisfies F, the following holds, where * is the natural join of all the relations in D:

* (R1(r), ..., Rm(r)) = r

Note: The word loss in lossless refers to loss of information, not to loss of tuples. In fact, for “loss of information” a better term is “addition of spurious information”

Example

R{A,B,C} decompose to {AB,AC} r A B C r1 A B r2 A C 1 1 1 1 1 1 1 1 2 1 1 2

r1 * r2=r

Example

R{A,B,C} decompose to {AB,AC} r A B C r1 A B r2 A C 1 1 4 1 1 1 4 1 2 3 1 2 1 3

r1 * r2=r

Properties of Relational Decompositions (6)

Lossless (Non-additive) Join Property of a Decomposition (cont.): Algorithm 8.1: Testing for Lossless Join Property

Input: A universal relation R, a decomposition D = {R1, R2, ..., Rm} of R, and a set F of functional dependencies.

1. Create an initial matrix S with one row i for each relation Ri in D, and one column j for each attribute Aj in R.

2. Set S(i,j):=bij for all matrix entries. (* each bij is a distinct symbol associated with indices (i,j) *).

3. For each row i representing relation schema Ri

{for each column j representing attribute Aj

{if (relation Ri includes attribute Aj) then set S(i,j):= aj;};};

(* each aj is a distinct symbol associated with index (j) *)

Properties of Relational Decompositions (7)

Lossless (Non-additive) Join Property of a Decomposition (cont.): Algorithm 8.1: Testing for Lossless Join Property (cont.)4. Repeat the following loop until a complete loop execution results in no

changes to S             {for each functional dependency X Y in F

{for all rows in S which have the same symbols in the columns corresponding to attributes in X

{make the symbols in each column that correspond to an attribute in Y be the same in all these rows as follows: if any of the rows has an “a” symbol for the column, set the other rows to that same “a” symbol in the column. If no “a” symbol exists for the attribute in any of the rows, choose one of the “b” symbols that appear in one of the rows for the attribute and set the other rows to that same “b” symbol in the column ;};};};

5. If a row is made up entirely of “a” symbols, then the decomposition has the lossless join property; otherwise it does not.

Properties of Relational Decompositions (8)

Lossless (nonadditive) join test for n-ary decompositions. (a) Case 1: Decomposition of EMP_PROJ into EMP_PROJ1 and EMP_LOCS fails test. (b) A decomposition of EMP_PROJ that has the lossless join property.

Properties of Relational Decompositions (8)

Lossless (nonadditive) join test for n-ary decompositions. (c) Case 2: Decomposition of EMP_PROJ into EMP, PROJECT, and WORKS_ON satisfies test.

Exercises

(1)R(ABCDE),F{AB->C,C->D,D->E} decompose to R1(ABC),R2(CD),R3(DE) is it lossless decomposition?

the answer is yes. (2) R(ABC),F{A->B,C->B} decomposition1{R1(AB),R2(AC)} decomposition2{R1(AB),R2(BC)} ARE THEY DECOMPOSITION? THE SECOND IS NOT.

Properties of Relational Decompositions (9)

Testing Binary Decompositions for Lossless Join Property: Binary Decomposition: decomposition of a relation R into two

relations. PROPERTY LJ1 (lossless join test for binary

decompositions): A decomposition D = {R1, R2} of R has the

lossless join property with respect to a set of functional dependencies F on R if and only if either

The f.d. ((R1 ∩ R2) (R1- R2)) is in F+, or

The f.d. ((R1 ∩ R2) (R2 - R1)) is in F+.

Algorithms for Relational Database Schema Design (2)

Algorithm 8.3: Relational Decomposition into BCNF with Lossless (non-additive) join property

Input: A universal relation R and a set of functional dependencies F on the attributes of R.

1. Set D := {R};

2. While there is a relation schema Q in D that is not in BCNF

do {

choose a relation schema Q in D that is not in BCNF;

           find a functional dependency X Y in Q that violates BCNF;

           replace Q in D by two relation schemas (Q - Y) and (X υ Y);

};

Assumption: No null values are allowed for the join attributes.

Algorithms for Relational Database Schema Design (3)

Algorithm 8.4 Relational Synthesis into 3NF with Dependency Preservation and Lossless (Non-Additive) Join Property

Input: A universal relation R and a set of functional dependencies F on the attributes of R.

1. Find a minimal cover G for F (Use Algorithm 10.2).2. For each left-hand-side X of a functional dependency that appears in G, cr

eate a relation schema in D with attributes {X υ {A1} υ {A2} ... υ {Ak}}, where X A1, X A2, ..., X –>Ak are the only dependencies in G with X as left-hand-side (X is the key of this relation).

3. If none of the relation schemas in D contains a key of R, then create one more relation schema in D that contains attributes that form a key of R. (Use Algorithm 11.4a to find the key of R)

R(FGHIJ) FD={F->I,J->I,I->G,GH->I,IH->F} decompose it to

3nf and preserve functional dependency and lossless connection.

HJ is the candidate key. (1):Fmin{F->I,J->I,I->G,GH->I,IH->F} (2):{R1(FI),R2(JI),R3(IG),R4(GHI),R5(IHF)} (3): ADD candidate key:

{R1(FI),R2(JI),R3(IG),R4(GHI),R5(IHF),R6(HJ)}

Algorithms for Relational Database Schema Design (4)

Algorithm 8.4a Finding a Key K for R Given a set F of Functional Dependencies

Input: A universal relation R and a set of functional dependencies F on the attributes of R.

1. Set K := R.

2. For each attribute A in K {

           compute (K - A)+ with respect to F;

If (K - A)+ contains all the attributes in R,

then set K := K - {A}; }

Chapter 10-74

Example 1 (Exercises 10.28)

Consider the following relation：A B C TUPLE#

10 b1 c1 #1

10 b2 c2 #2

11 b4 c1 #3

12 b3 c4 #4

13 b4 c1 #5

14 b3 c4 #6(A).Give the previous extension (state), which dependencies may hold

in the above reltion? If the dependency cannot hold, explain why by specifying the tuples that cause the violation.

(B).Does the above relation have a potential candidate key?If it does, what is it?If it does not, why not?

Chapter 10-75

Example 1- Function Dependency

1. A→B 不成立，給定某一Ａ值，則出來的 B 值會重覆 ( 多個 ≧ 2) 　　 如： A=10 時 , B=b1,b2 Ex: tuple 1,2 A→C 不成立，給定某一Ａ值，則出來的 B 值會重覆 ( 多個 ≧ 2) 　　 如： A=10 時 , C=c1,c2 Ex: tuple 1,2 B→C 成立，給定某一Ｂ值，出來的 C 值會不重覆 (C 值唯一 ) 　　 如： B=b4 時 , C=c1 ( 唯一值 ) Ex: tuple 3,5 (or Ex: tuple 4,6) C→B 不成立，給定某一Ｃ值，則出來的Ｂ值會重覆 ( 多個 ≧ 2) 　　 如： C=C1 時 , B=b1,b4 Ex: tuple 1,3 B→A 不成立，給定某一Ｂ值，則出來的 A 值會重覆 ( 多個 ≧ 2) 　　 如： B=b3 時 , A=12,14 Ex: tuple 4,6 C→A 不成立，給定某一Ｃ值，則出來的 A 值會重覆 ( 多個 ≧ 2) 　　 如： C=c1 時 , A=10,11,13 Ex: tuple 1,3,5 AB →C 成立 原因略 AC →B 成立 BC →A 不成立

2. Since none of the single attributes is a key( whose closure includes all the attributes ),

The key will be AB,AC whose closure includes all the attributes ∵ BC →A is not hold, ∵ whose closure doesn’t includes all the attributes

A B C TUPLE#

10 b1 c1 #1

10 b2 c2 #2

11 b4 c1 #3

12 b3 c4 #4

13 b4 c1 #5

14 b3 c4 #6

Chapter 10-76

Example 2 (Exercises 10.26)

Relation R = {A, B, C, D, E, F, G, H, I,J} Functional dependencies F = { {A, B} -> {C},

{A} -> {D, E}, {B} -> {F}, {F} -> {G, H}, {D} -> {I, J} }.

1.What is the key for R? 2.Decompose R into 2NF relations, 3.Then Decompose R into 3NF relations.

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Chapter 10-77

Example 2 – Function Dependency1.A minimal set of attributes whose closure includes all the attributes in R is a

key. {A, B}+ = R, {A, B} is key.

2.First, identify partial dependencies that violate 2NF only 1NF. These are attributes that are functionally dependent on either parts of the key, {A} or {B}, alone. We can calculate the closures {A}+ and {B}+ to determine partially dependent attributes:{A}+ = {A, D, E, I, J}. Hence {A} -> {D, E, I, J} ∵{A} -> {D, E}, {D} -> {I, J} {B}+ = {B, F, G, H}, hence {B} -> {F, G, H} ∵{B} -> {F}, {F} -> {G, H}

Second, To normalize into 2NF, We remove the attributes that are functionally dependent on part of the key (A or B) from R and place them in separate relations R1 and R2, but also remains in the original relation is R3:

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Chapter 10-78

Example 2- Decomposition

R1 = {A, D, E, I, J}, R2 = {B, F, G, H}, R3 = {A, B, C}3.Third, look for transitive dependencies in R1, R2, R3. (dot line)

The relation R1 has the transitive dependency {A} -> {D, E} -> {I, J} ∵{A} -> {D, E}, {D} -> {I, J}we remove the transitively dependent attributes {D} -> {I, J} from R1 into a relation R11. The remaining attributes are kept in a relation R12. R11 = {D, I, J}, R12 = {A, D, E}The relation R2 has the transitive dependency {B} -> {F} -> {G, H}: ∵{B} -> {F}, {F} -> {G, H} decomposed into R21 = {F, G, H}, R22 = {B, F}

The final set of relations in 3NF are {R11, R12, R21, R22, R3}R11 = {D, I, J}, R12 = {A, D, E}R21 = {F, G, H}, R22 = {B, F} R3 = {A, B, C}

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