1. Normal forms First Normal Form (1NF) Second Normal Form (2NF)
Third Normal Form (3NF)
Boyce/Codd Normal Form (BCNF)
Fourth Normal Form (4NF)
Fifth Normal Form (5NF)

2. First normal form (1NF)
The only attribute values permitted are single atomic (indivisible)
Not allow a set, list, bag, etc.
considered to be part of the formal definition of a relations

3. Various functional dependencies
Prime and nonprime attribute
an attribute of a relation schema R is called a prime attribute of R if it is a member of any candidate key of R
an attribute is called nonprime if it is not a prime attribute, i.e. not a member of any candidate key of R
Full functional dependency
Y is said to be fully dependent on X if X  Y and Z  Y for any X  Z
Y is fully dependent on X if and only if
Y is functionally dependent on X, and -
not functionally dependent on any proper subset of X
Partial functional dependency
Y is said to be partially dependent on X if some attribute can be removed from X and the dependency still holds
Transitive functional dependency

4. Second Normal Form (2NF)
A relation schema R is in 2NF if it is in 1NF and every nonprime attribute A in R is fully functionally dependent on every key of R
R(SSN, pNumber, hours, eName, pName, pLocation) SSN pNumber  hours SSN  eName pNumber  pName pLocation decomposed into R1(SSN, pNumber, hours) R2(SSN, eName) R3(pNumber, pName, pLocation)

5. Example for 1NF and 2NF
FIRST(S#, P#, Status, City, Qty) S# P#  Status City Qty S#  City Status City  Status
Anomalies
insertion: cannot record City for a supplier until he supplies something
modification: City for a supplier appears many times
deletion: deletion of last tuple for S# lose its City
FIRST is not in 2NF, so decompose into SECOND(S#, Status, City) SP(S#, P#, Qty) S#  City Status City  Status S# P#  Qty

6. Third Normal Form (3NF)
A relation schema is in 3NF if it is in 2NF and no nonprime attribute of R is transitively dependent on any key
Alternatively, a relation schema R is in 3NF if whenever a nontrivial functional dependency X -->A holds in R, then either (a) X is a super key of R or (b) A is a prime attribute of R
EMP_DEPT(SSN, Ename, Bdate, Addr, D#, Dname, Dmgrssn) FDs: SSN  Ename Bdate Addr D# D#  Dname Dmgrssn Since Dname and Dmgrssn are transitively dependent on D# (not in 3NF) ED1(SSN, Ename, Bdate, Addr, D#) ED2(D#, Dname, Dmgrssn)

7. Example for 2NF and 3NF
SECOND(S#, Status, City) SP(S#, P#, Qty) S#  City Status City  Status
Anomalies
insertion: cannot record new Status for a city without S#
modification : Status for a City appears in many tuples
deletion : delete only the second tuple for a particular City
SECOND is not in 3NF, so decompose into SC(S#,City) CS(City, Status)

8. Boyce/Codd Normal Form (BCNF)
A relation schema R is in BCNF if whenever a nontrivial functional dependency X  A holds in R, then X is a super key of R
Difference with 3NF
Drop the second condition in 3NF that allows A to be prime if X is not a superkey

9. 3NF and BCNF example
MovieStudio(title, year, length, filmType, studioName, studioAddr) title year  length filmType studioName studioName  studioAddr
Key: {title, year} Hence, MovieStudio is not 3NF
Decompose into MovieStudio1(title, year, length, filmType, studioName) MovieStudio2(studioName, studioAddr) Then we get a schema in BCNF.

10. All binary relations are in BCNF
Let A and B are all attributes
Consider all possible cases, here there are totally 4 cases
no nontrivial FD at all:
{A, B} is a key, so in BCNF
A  B holds, B  A does not hold
{A} is a key, so in BCNF
B  A holds, A  B does not hold: similarly
A  B holds, B  A holds
{A} and {B} are keys, so in BCNF

11. Multivalued dependency example
EMP(eName, pName, depName) Smith {X,Y} {John, Anna}
Must repeat every combination due to 1NF
Two independent one-many relationships are mixed in the same relation
eName -->> pName eName -->> depName

12. Multivalued dependency definition
Let X, Y be sets of attributes in R Let Z be compliment of X  Y
Multivalued dependency (MVD) X -->> Y [X multidetermines Y] holds in R if and only if each X-value in R is associated with a set of Y-values in a way that does not depend on Z-values
MVD and FD-MVD rules
Complementation
If X -->> Y, then X -->> T – X – Y where Ts are all attributes of R not among the A’s and B’s
Augmentation
If X -->> Y and V  W, XW -->> YV
Transitivity
If X -->> Y and Y -->>Z, then X -->> Z – Y
If X  Y, then X -->> Y (i.e. an FD is a special case of an MVD)

13. Fourth normal form (4NF)
An MVD A1A2 … An -->> B1B2 … Bm for a relation R is trivial if:
A’s  B’s or
A’s  B’s are all attributes of R
A relation R is in 4NF if whenever A1A2 … An -->> B1B2 … Bm is a nontrivial MVD, {A1, A2, …, An} is a superkey

14. 4NF example
Star(name, street, city, title, year) name -->> street city
Star relation is not in 4NF, hence decompose into Star1(name, street, city) Star2(name, title, year)
Note that both new relations are in 4NF

15. Relationship among normal forms
All relation schema
1NF
2NF
3NF
BCNF
4NF
5NF
In practice, it is best to have relation schemas in BCNF or in 3NF