Relational Normalization Theory. Dr. Philip Cannata

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1 Relational Normalization Theory Dr. Philip Cannata 1

Designing Databases Using Normalization On No!

F+, A+, FD Preservation, Lossless Join See also

2 Designing Databases Using Normalization On No!!! There s another way to design a good set of tables besides Conceptual and Logical Modeling. Normalization FDs, Keys, 1NF, 2NF, 3NF, BCNF, MVDs, F+, A+, FD Preservation, Lossless Join See also Dr. Philip Cannata 2

3 EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC SMITH CLERK DEC RESEARCH DALLAS 7499 ALLEN SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB SALES CHICAGO 7566 JONES MANAGER APR RESEARCH DALLAS 7654 MARTIN SALESMAN SEP SALES CHICAGO 7698 BLAKE MANAGER MAY SALES CHICAGO 7782 CLARK MANAGER JUN ACCOUNTING NEW YORK 7788 SCOTT ANALYST DEC RESEARCH DALLAS 7839 KING none 17-NOV ACCOUNTING NEW YORK 7844 TURNER SALESMAN SEP SALES CHICAGO 7876 ADAMS CLERK JAN RESEARCH DALLAS 7900 JAMES CLERK DEC SALES CHICAGO 7902 FORD ANALYST DEC RESEARCH DALLAS 7934 MILLER CLERK JAN ACCOUNTING NEW YORK Insertion Anomalies What s wrong with this table? Can t insert information about Department 40 unless it has an employee. So where do you store information about Department 40? Deletion Anomalies If you delete EMPNOs 7782, 7839, and 7934, you lose all information about Department 10. Update Anomalies A Badly Designed Table* *This table was create with the following SQL: create table emp_dept as (select e.*, dname, loc from emp e, dept d where e.deptno = d.deptno) If you change CLARK s DNAME to SALES and leave his DEPTNO=10, you need to make sure the DNAME for DEPT 10 is changed in every tuple. Dr. Philip Cannata 3

4 What does No Joins Mean in MongoDB? {"company" : [ { "name":"accounting","identifier":"10","location":"new YORK", "employees":[{ "name":"clark","job":"manager","salary":2450, "manager":{ "name":"king","salary": 5000,"hiredate":" "},"hiredate":" "}, { "name":"king","job":"president","salary":5000,"manager":"","hiredate":" "} ]}, { "name":"operations","identifier":"40","location":"boston", "employees":[]}, { "name":"research","identifier":"20","location":"dallas", "employees":[{ "name":"adams","job":"clerk","salary":1100, "manager":{ "name":"scott","salary": 3000,"hiredate":" "},"hiredate":" "}, { "name":"ford","job":"analyst","salary":3000, "manager":{ "name":"jones","salary":2975,"hiredate":" "},"hiredate":" "}, { "name":"jones","job":"manager","salary":2975,"manager":{ "name":"king","salary":5000,"hiredate":" "},"hiredate":" "}, { "name":"scott","job":"analyst","salary":3000,"manager":{ "name":"jones","salary":2975,"hiredate":" "},"hiredate":" "}, { "name":"smith","job":"clerk","salary":800, "manager":{ "name":"ford","salary":3000,"hiredate":" "},"hiredate":" "} ]}, { "name":"sales","identifier":"30","location":"chicago", "employees":[{ "name":"allen","job":"salesman","salary":1600, "manager":{ "name":"blake","salary": 2850,"hiredate":" "},"hiredate":" "}, { "name":"blake","job":"manager","salary":2850, "manager":{ "name":"king","salary":5000,"hiredate":" "},"hiredate":" "}, { "name":"james","job":"clerk","salary":950, "manager":{ "name":"blake","salary":2850,"hiredate":" "},"hiredate":" "}, { "name":"martin","job":"salesman","salary":1250,"manager":{ "name":"blake","salary":2850,"hiredate":" "},"hiredate":" "}, { "name":"turner","job":"salesman","salary":1500,"manager":{ "name":"blake","salary":2850,"hiredate":" "},"hiredate":" "}, { "name":"ward","job":"salesman","salary":1250, "manager":{ "name":"blake","salary":2850,"hiredate":" "},"hiredate":" "} ]} ]} Is This a Good Thing? Dr. Philip Cannata 4

5 If A Side Discussion of Formal Systems (This and the next slides with green background are optional) If a -> a b c d e f d -> e f Then a ->a b c d a b -> a b c d e f a -> c d b -> e f Then a b -> a b d -> e f a -> c d b -> e f Dr. Philip Cannata 5

6 A Badly Designed Table* EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC SMITH CLERK DEC RESEARCH DALLAS 7499 ALLEN SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB SALES CHICAGO 7566 JONES MANAGER APR RESEARCH DALLAS 7654 MARTIN SALESMAN SEP SALES CHICAGO 7698 BLAKE MANAGER MAY SALES CHICAGO 7782 CLARK MANAGER JUN ACCOUNTING NEW YORK 7788 SCOTT ANALYST DEC RESEARCH DALLAS 7839 KING none 17-NOV ACCOUNTING NEW YORK 7844 TURNER SALESMAN SEP SALES CHICAGO 7876 ADAMS CLERK JAN RESEARCH DALLAS 7900 JAMES CLERK DEC SALES CHICAGO 7902 FORD ANALYST DEC RESEARCH DALLAS 7934 MILLER CLERK JAN ACCOUNTING NEW YORK *This table was create with the following SQL: create table emp_dept as (select e.*, dname, loc from emp e, dept d where e.deptno = d.deptno) In this table EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC and DEPTNO -> DNAME LOC -> stands for Funcionally Determines which we will discuss later. Dr. Philip Cannata 6

7 If Apply the Simple Formal System R = emp_dept Relation (Table) R EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC DEPTNO -> DNAME LOC Then R1 = emp Relation (Table) R1 R2 EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO DEPTNO -> DNAME LOC R2 = dept Relation (Table) So, using the Simple Formal System, we decomposed a bad table R (which is said to not be in 3NF) into a set of tables R1 and R2 that are in 3NF. (The formal definition of 3NF will be given later.) Dr. Philip Cannata 7

At the turn of the twentieth Century, mathematicians asked Why wouldn t it be true that all theorems in arithmetic can be proven by a formal system.

8 At the turn of the twentieth Century, mathematicians asked Why wouldn t it be true that all theorems in arithmetic can be proven by a formal system. (This question was implicit in the actual problem put forth by David Hilbert in 1900 Prove that the axioms of arithmetic are consistent. This lead to 30 years of work and a three volume publication - Principia Mathematica Dr. Philip Cannata 8

In 1930, Kurt Gödel answered the Why wouldn t it be true question with his First Incompleteness

First Incompleteness Theorem: There are true statements in mathematics that no formal system can

Second Incompleteness Theorem: No formal system can prove its own consistency.

9 In 1930, Kurt Gödel answered the Why wouldn t it be true question with his First Incompleteness Theorem In my opinion, one of the two* greatest discoveries of humankind. First Incompleteness Theorem: There are true statements in mathematics that no formal system can prove true and there are false statements in mathematics that any formal system will prove true. Second Incompleteness Theorem: No formal system can prove its own consistency. *The second is John Stewart Bell s inequality Theorem, see the book The Age of Entanglement. Dr. Philip Cannata See also Gödel's papers on the class website 9

10 Dr. Philip Cannata 10

11 Good Books to Have for a Happy Life J From Frege to Gödel: My Favorite all books books I ve ever read in over 65 years. Dr. Philip Cannata 11

12 Functional Dependencies Informal definition of a Functional Dependency Require that the value for a certain set of attributes uniquely determines the value for another set of attributes. A functional dependency is a generalization of the notion of a key. Formal definition of a Functional Dependency Let R be a relation and α R and β R The functional dependency α β holds on R if and only if for any tuples t 1 and t 2 that agree on the attributes α, they also agree on the attributes β. That is, t 1 [α] = t 2 [α] t 1 [β ] = t 2 [β ] Dr. Philip Cannata 12

13 Keys Keys Superkey - K is a superkey for relation R if and only if K R Candidate Key - K is a candidate key for R if and only if K R (i.e., K is a super key), and for no α K, α R (i.e., K is minimal with respect to the number of attributes of which it is composed) Prime attributes - attributes that belong to any candidate key Primary Key Pick one from the candidate keys Dr. Philip Cannata 13

14 Armstrong s Axioms Armstrong s Axioms: if β α, then α β (reflexivity) (e.g., if α = AB then AB A and AB B) (these are the trivial dependencies) if α β, then γ α β (augmentation) and γ α γ β if α β, and β γ, then α γ (transitivity) Dr. Philip Cannata 14

15 Additional Rules Additional rules implied by Armstrong s Axioms: if α β holds and α γ holds, then α β γ holds (union) if α β γ holds, then α β holds and α γ holds (decomposition) if α β holds and γ β δ holds, then α γ δ holds (pseudotransitivity) Dr. Philip Cannata 15

16 R = (A, B, C, G, H, I) F = { A B A C CG H CG I B H} Example of Finding More FDs some members of F + A H by transitivity from A B and B H AG I by augmenting A C with G, to get AG CG and then transitivity with CG I CG HI from CG H and CG I : union rule can be inferred from definition of functional dependencies, or Augmentation of CG I to infer CG CGI, augmentation of CG H to infer CGI HI, and then transitivity Dr. Philip Cannata 16

17 3NF 3NF - Relation R is in 3NF if: R is in 2NF and No nonprime attribute functionally determines any other nonprime attribute Dr. Philip Cannata 17

18 Designing Databases Using Normalization 3NF A good way to remember this. If you have a SINGLE table that is a mapping from the following object model (instead of the normal two tables), A * * B the single table will not be in 3NF. Dr. Philip Cannata 18

19 Designing Databases Using Normalization The problem is that some non-key attribute(s) are functionally dependent on a non-key attribute (this is not allowed for 3NF). Decomposition into 3NF A B C Convert to B C + A B A à B C B à C Offending Functional Dependency, make it its own table B à C A à B This will be a Lossless-Join decomposition, see the next 2 pages for a description of a Lossless-Join decomposition. Dr. Philip Cannata 19

20 Designing Databases Using Normalization Lossless-Join Decomposition Example of Non Lossless-Join Decomposition Decompose empdept (create table empdept as select * from emp natural join dept) into 2 tables: create table emp_dept1 as select empno, ename, job, mgr, hiredate, sal, comm from emp_dept create table emp_dept2 as select job, deptno, dname, loc from emp_dept Then try to recreate empdept by issuing the following query select * from emp_dept1 natural join emp_dept2 Dr. Philip Cannata 20

21 Designing Databases Using Normalization Lossless-Join Decomposition Test for Lossless-Join Decomposition Let R be a relation that you want to decompose into R1 and R2. The decomposition is Lossless if R1 R2 à R1 or R1 R2 à R2 In other words, if R1 R2 forms a superkey of either R1 or R2 Dr. Philip Cannata 21

22 A Badly Designed (Non-2NF) Table* EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC SMITH CLERK DEC RESEARCH DALLAS 7369 SMITH CLERK DEC SALES CHICAGO 7499 ALLEN SALESMAN FEB ACCOUNTING NEW YORK 7499 ALLEN SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB ACCOUNTING NEW YORK 7566 JONES MANAGER APR SALES CHICAGO 7566 JONES MANAGER APR RESEARCH DALLAS 7654 MARTIN SALESMAN SEP SALES CHICAGO 7654 MARTIN SALESMAN SEP ACCOUNTING NEW YORK 7698 BLAKE MANAGER MAY SALES CHICAGO 7698 BLAKE MANAGER MAY ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN RESEARCH DALLAS 7788 SCOTT ANALYST DEC RESEARCH DALLAS 7788 SCOTT ANALYST DEC SALES CHICAGO 7839 KING none 17-NOV ACCOUNTING NEW YORK 7839 KING none 17-NOV RESEARCH DALLAS 7844 TURNER SALESMAN SEP SALES CHICAGO 7844 TURNER SALESMAN SEP ACCOUNTING NEW YORK 7876 ADAMS CLERK JAN RESEARCH DALLAS 7876 ADAMS CLERK JAN SALES CHICAGO 7900 JAMES CLERK DEC SALES CHICAGO 7900 JAMES CLERK DEC ACCOUNTING NEW YORK 7902 FORD ANALYST DEC RESEARCH DALLAS 7902 FORD ANALYST DEC SALES CHICAGO 7934 MILLER CLERK JAN RESEARCH DALLAS 7934 MILLER CLERK JAN ACCOUNTING NEW YORK *This table was create with the following SQL: insert into emp_dept (select t.*, dname, loc from (select EMPNO, ENAME, JOB, MGR,HIREDATE, SAL, COMM, case when deptno = 10 then 20 when deptno = 20 then 30 when deptno = 30 then 10 Dr. Philip Cannata end deptno from emp) t, dept d where t.deptno = d.deptno) 22

23 The Badly Designed Table EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC SMITH CLERK DEC RESEARCH DALLAS 7369 SMITH CLERK DEC SALES CHICAGO 7499 ALLEN SALESMAN FEB ACCOUNTING NEW YORK 7499 ALLEN SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB ACCOUNTING NEW YORK 7566 JONES MANAGER APR SALES CHICAGO 7566 JONES MANAGER APR RESEARCH DALLAS 7654 MARTIN SALESMAN SEP SALES CHICAGO 7654 MARTIN SALESMAN SEP ACCOUNTING NEW YORK 7698 BLAKE MANAGER MAY SALES CHICAGO 7698 BLAKE MANAGER MAY ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN RESEARCH DALLAS 7788 SCOTT ANALYST DEC RESEARCH DALLAS 7788 SCOTT ANALYST DEC SALES CHICAGO 7839 KING none 17-NOV ACCOUNTING NEW YORK 7839 KING none 17-NOV RESEARCH DALLAS 7844 TURNER SALESMAN SEP SALES CHICAGO 7844 TURNER SALESMAN SEP ACCOUNTING NEW YORK 7876 ADAMS CLERK JAN RESEARCH DALLAS 7876 ADAMS CLERK JAN SALES CHICAGO 7900 JAMES CLERK DEC SALES CHICAGO 7900 JAMES CLERK DEC ACCOUNTING NEW YORK 7902 FORD ANALYST DEC RESEARCH DALLAS 7902 FORD ANALYST DEC SALES CHICAGO 7934 MILLER CLERK JAN RESEARCH DALLAS 7934 MILLER CLERK JAN ACCOUNTING NEW YORK Insertion Anomalies What s wrong with this table? Can t insert information about Department 40 unless it has an employee. So where do you store information about Department 40? Deletion Anomalies If you delete EMPNOs in department 10, you lose all information about Department 10. Update Anomalies If change CLARK s DNAME to SALES where DEPTNO = 10 and leave his DEPTNO=10, you need to make sure the DNAME for DEPT 10 is changed in every tuple. Dr. Philip Cannata 23

24 The Badly Designed Table EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC SMITH CLERK DEC RESEARCH DALLAS 7369 SMITH CLERK DEC SALES CHICAGO 7499 ALLEN SALESMAN FEB ACCOUNTING NEW YORK 7499 ALLEN SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB SALES CHICAGO 7521 WARD SALESMAN FEB ACCOUNTING NEW YORK 7566 JONES MANAGER APR SALES CHICAGO 7566 JONES MANAGER APR RESEARCH DALLAS 7654 MARTIN SALESMAN SEP SALES CHICAGO 7654 MARTIN SALESMAN SEP ACCOUNTING NEW YORK 7698 BLAKE MANAGER MAY SALES CHICAGO 7698 BLAKE MANAGER MAY ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN ACCOUNTING NEW YORK 7782 CLARK MANAGER JUN RESEARCH DALLAS 7788 SCOTT ANALYST DEC RESEARCH DALLAS 7788 SCOTT ANALYST DEC SALES CHICAGO 7839 KING none 17-NOV ACCOUNTING NEW YORK 7839 KING none 17-NOV RESEARCH DALLAS 7844 TURNER SALESMAN SEP SALES CHICAGO 7844 TURNER SALESMAN SEP ACCOUNTING NEW YORK 7876 ADAMS CLERK JAN RESEARCH DALLAS 7876 ADAMS CLERK JAN SALES CHICAGO 7900 JAMES CLERK DEC SALES CHICAGO 7900 JAMES CLERK DEC ACCOUNTING NEW YORK 7902 FORD ANALYST DEC RESEARCH DALLAS 7902 FORD ANALYST DEC SALES CHICAGO 7934 MILLER CLERK JAN RESEARCH DALLAS 7934 MILLER CLERK JAN ACCOUNTING NEW YORK In this table EMPNO DEPTNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO -> DNAME LOC Dr. Philip Cannata 24

25 A Simple Formal System If a -> a b c d e f d -> e f Then a ->a b c d If a b -> a b c d e f a -> c d b -> e f Then a b -> a b d -> e f a -> c d b -> e f Dr. Philip Cannata 25

26 If Apply the Simple Formal System R = emp_dept Relation (Table) EMPNO DEPTNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO DNAME LOC R EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO -> DNAME LOC Then R1 R1 R2 R3 EMPNO DEPTNO -> EMPNO DEPTNO R2 = emp Relation (Table) without DEPTNO FK EMPNO -> ENAME JOB MGR HIREDATE SAL COMM DEPTNO -> DNAME LOC R3 = dept Relation (Table) So, using the Simple Formal System, we decomposed a bad table R (which is said to not be in 2NF) into a set of tables R1, R2 and R that are in 2NF. (The formal definition of 2NF will be given later.) Dr. Philip Cannata 26

27 2NF - Relation R is in 2NF if: R is in 1NF and 2NF there is no nonprime attribute (i.e., an attribute that s not part of any candidate key) that is dependent on any part of the candidate key (no nonprime attribute is functionally determined by a prime attribute) in other words, each nonprime attribute in relation R is fully dependent upon every candidate key Note: If relation R has no compound keys, R is automatically in 2NF Dr. Philip Cannata 27

28 Designing Databases Using Normalization 2NF A good way to remember this. If you have a SINGLE table that is a mapping from the following object model (instead of the normal three tables), A * * B the single table will not be in 2NF. Dr. Philip Cannata 28

29 Designing Databases Using Normalization The problem is that some non-key attributes are functionally dependent on PART of the key (this is not allowed for 2NF). Decomposition into 2NF A B C D A B à C D A à D Convert to Offending Functional Dependency, make it B C A D B B à C A B à D A à D B à C its own table B C A D Convert to A B This will be a Lossless-Join decomposition. B à C A à D AB à AB Dr. Philip Cannata 29

30 1NF - Relation R is in 1NF if: 1NF All data stored at the intersection of a row (tuple) and column (attribute) must be atomic (i.e., it can t be decomposed into multiple values). See next pages for examples. A table (Relation) must not contain any repeating columns (attributes). See next pages for examples. Dr. Philip Cannata 30

31 1NF All values must be Atomic Badly Designed Table EMPNO ENAME JOB MGR HIREDATE SAL COMM DEPTNO SMITH CLERK, Another Job DEC ALLEN SALESMAN, Another Job FEB WARD SALESMAN, Another Job FEB JONES MANAGER, Another Job APR MARTIN SALESMAN, Another Job SEP BLAKE MANAGER, Another Job MAY CLARK MANAGER, Another Job JUN SCOTT ANALYST, Another Job DEC KING none, Another Job 17-NOV TURNER SALESMAN, Another Job SEP ADAMS CLERK, Another Job JAN JAMES CLERK, Another Job DEC FORD ANALYST, Another Job DEC MILLER CLERK, Another Job JAN The Fix Or, in some cases, create a table with a type column that represents a hierarchy as in the Business Contracts Contact table à. Dr. Philip Cannata 31

Another Job 7839 02-APR-81 2975 20 7654 MARTIN SALESMAN Another Job 7698 28-SEP-81 1250 1400 30 7698 BLAKE MANAGER Another Job 7839 01-MAY-81 2850 30 7782 CLARK MANAGER Another Job 7839 09-JUN-81

32 1NF No Repeating Columns (Attributes) Badly Designed Table EMPNO ENAME JOB JOB2 MGR HIREDATE SAL COMM DEPTNO SMITH CLERK Another Job DEC ALLEN SALESMAN Another Job FEB WARD SALESMAN Another Job FEB JONES MANAGER Another Job APR MARTIN SALESMAN Another Job SEP BLAKE MANAGER Another Job MAY CLARK MANAGER Another Job JUN SCOTT ANALYST Another Job DEC KING none Another Job 17-NOV TURNER SALESMAN Another Job SEP ADAMS CLERK Another Job JAN JAMES CLERK Another Job DEC FORD ANALYST Another Job DEC MILLER CLERK Another Job JAN The Fix Or, in some cases, create a table with a type column that represents a hierarchy as in the Business Contracts Contact table à. Dr. Philip Cannata 32

33 Try This Work from left to right à Given the Relation R=ABCDEF that s in 1NF and the following set F of Functional Dependencies, AàBC DàEF EàF find a set of Relations that are in 3NF. Armstrong s Axioms: You might need the following to find the Primary Key of R. if β α, then α β (reflexivity) (e.g., if α = AB then AB A and AB B) (these are the trivial dependencies) if α β, then γ α β (augmentation) and γ α γ β if α β, and β γ, then α γ (transitivity) Additional rules implied by Armstrong s Axioms: if α β holds and α γ holds, then α β γ holds (union) if α β γ holds, then α β holds and α γ holds (decomposition) if α β holds and γ β δ holds, then α γ δ holds (pseudotransitivity) Dr. Philip Cannata 33

34 Another Example Dr. Philip Cannata 34

35 Second Normal Form (2NF) Conversion Results Dr. Philip Cannata 35

36 Third Normal Form (3NF) Conversion Results Dr. Philip Cannata 36

37 Formal Treatment of Normalization Dr. Philip Cannata 37

38 Designing Databases Using Normalization Finding Keys (Armstrong s Axioms) Armstrong s Axioms: if β α, then α β (reflexivity) (e.g., if α = AB then AB A and AB B) (these are the trivial dependencies) if α β, then γ α β (augmentation) and γ α γ β if α β, and β γ, then α γ (transitivity) Dr. Philip Cannata 38

39 Designing Databases Using Normalization Finding Keys (additional rules) Additional rules implied by Armstrong s Axioms: if α β holds and α γ holds, then α β γ holds (union) if α β γ holds, then α β holds and α γ holds (decomposition) if α β holds and γ β δ holds, then α γ δ holds (pseudotransitivity) Dr. Philip Cannata 39

40 Finding Keys (F + ) Procedure for Computing F + To compute the closure of a set of functional dependencies F: F + = F repeat for each functional dependency f in F + apply reflexivity and augmentation rules on f add the resulting functional dependencies to F + for each pair of functional dependencies f 1 and f 2 in F + if f 1 and f 2 can be combined using transitivity then add the resulting functional dependency to F + until F + does not change any further Dr. Philip Cannata 40

41 R = (A, B, C, G, H, I) F = { A B A C CG H CG I B H} Example some members of F + A H by transitivity from A B and B H AG I by augmenting A C with G, to get AG CG and then transitivity with CG I CG HI from CG H and CG I : union rule can be inferred from definition of functional dependencies, or Augmentation of CG I to infer CG CGI, augmentation of CG H to infer CGI HI, and then transitivity Dr. Philip Cannata 41

42 Designing Databases Using Normalization Finding Keys (a + ) Closure of a set of attributes: Given a set of attributes a, define the closure of a under F (denoted by a + ) as the set of attributes that are functionally determined by a under F: if a β is in F + then β a + Algorithm to compute a +, the closure of a under F result := a; while (changes to result) do for each β γ in F do begin if β result then result := result γ end Dr. Philip Cannata 42

43 Designing Databases Using Normalization Finding Keys α is a superkey key if α + contains all attributes of R. It is also a candidate key if no subset of α is a candidate key. Dr. Philip Cannata 43

44 Designing Databases Using Normalization Finding Keys - Example R = (A, B, C, G, H, I) F = { A B A C CG H CG I B H } After trying A +, B +, C +, G +, H +, I +, (AB) +, (AC) + with no success, (AG) + R 1. result = AG 2. result = ABCG (from A C and A B) 3. result = ABCGH (from CG H) 4. result = ABCGHI (CG I) AG a superkey because: AG R AG is a candidate key because: A + does not R and G + does not R Dr. Philip Cannata 44

45 Designing Databases Using Normalization Dependency Preservation Decomposition Example of Non Dependency Preservation Decomposition Decompose empdept (create table empdept as select * from emp natural join dept) into 2 tables: create table empdept3 as select empno, ename, job, mgr, hiredate, sal, comm, deptno from empdept create table empdept4 as select distinct empno, dname, loc from empdept No Lossless-Join Problem but a test for the FDs deptno à dname and deptno à loc requires a join. Dr. Philip Cannata 45

46 Designing Databases Using Normalization Dependency Preservation Decomposition Test for Dependency Preservation Decomposition Let R be a relation that you want to decompose into R1, R2 Rn and Let each relation have the following Functional Dependencies R has F, R1 has F1, R2 has F2 and Rn has Fn The decomposition has Dependency Preservation if (F1 F2 Fn) + = F + For our example F is empno à R and deptno à deptno, dname, loc F1 is empno à R1 F2 is empno à R2 (F1 F2) + does not contain deptno à deptno, dname, loc Dr. Philip Cannata 46

47 Designing Databases Using Normalization 2NF - Relation R is in 2NF if: R is in 1NF and no nonprime attribute (i.e., an attribute that s not part of any candidate key) is partially dependent on any candidate key (no nonprime attribute is functionally determined by a prime attribute) in other words, each nonprime attribute in relation R is fully dependent upon every candidate key Note: If relation R has no compound keys, R is automatically in 2NF Dr. Philip Cannata 47

48 Designing Databases Using Normalization 3NF - Relation R is in 3NF if: R is in 2NF and No nonprime attribute functionally determines any other nonprime attribute Dr. Philip Cannata 48

49 Designing Databases Using Normalization BCNF - Relation R is in BCNF if: R is in 1NF and All functional dependencies must either be trivial or, their left hand sides must be a superkeys Note: This means all nonprime attributes must be fully dependent on every key (2NF) no nonprime attribute functionally determines any other nonprime attribute (3NF) (i.e. all BCNF Relations are in 3NF but not all 3NF Relations are in BCNF) Dr. Philip Cannata 49

50 Designing Databases Using Normalization BCNF Decomposition Algorithm result := {R}; done := false; compute F + ; while (not done) do if (there is a R i in result that is not in BCNF) then begin let α β be a nontrivial functional dependency that holds on R i such that α R i is not in F + (i.e., α is not a super key), and α β = ; result := { (result R i ) (R i β) } { α β } end else done := true; Note: each R i is in BCNF, and decomposition is Lossless-Join but not always Dependency Preserving Dr. Philip Cannata 50

51 A Dependency Diagram Dr. Philip Cannata 51

52 Second Normal Form (2NF) Conversion Results Dr. Philip Cannata 52

53 Third Normal Form (3NF) Conversion Results Dr. Philip Cannata 53

54 A Table That is in 3NF but not in BCNF Dr. Philip Cannata 54

55 Decomposition to BCNF Dr. Philip Cannata 55

56 Sample Data for a BCNF Conversion Dr. Philip Cannata 56

57 Another BCNF Decomposition Dr. Philip Cannata 57

58 Fourth Normal Form (4NF) Table is in fourth normal form (4NF) when both of the following are true: It is in 3NF Has no multiple sets of multivalued dependencies 4NF is largely academic if tables conform to following two rules: All attributes must be dependent on primary key, but independent of each other No row contains two or more multivalued facts about an entity Dr. Philip Cannata 58

59 Fourth Normal Form (4NF) (continued) Dr. Philip Cannata 59

60 Fourth Normal Form (4NF) (continued) Dr. Philip Cannata 60

CH-15 SIMPLE QUERY AND GROUPING

SQL SELECT STATEMENT. ASHOK GARG CH-15 SIMPLE QUERY AND GROUPING The SELECT statement is used to retrieve information from a table. Syntax: SELECT FROM [WHERE ] [GROUP