1NF = an unordered table of unordered atomic columns with no repeat rows
2NF = 1NF + all nonkey columns are fully functionally dependent on the PK
3NF = 2NF + all nonkey columns are mutually independent
BCNF = 3NF + every determinant (column on which some other column depends) is a candidate key (rowwise unique column combination, a column or column combo with no nulls & no dupes)
4NF = 3NF + all multivalued dependencies (MVDs) but one are projected to new tables (MVD = cols B & C depend on A but C does not depend on B).
5NF = 4NF + no cyclic dependencies ("join dependencies", JD).
Suppose you project a table with cyclic dependencies to N new tables, ie keep doing projections till there are no more non-key dependencies. N-decomposable means that having done these projections, recomposing the original table from the children gets you exactly the original table.
Another way of presenting it: cyclic dependency or JD exists in a table if for any i and j, if (Ai, Bi, Cj), eg A=Imre B=Izzy C=Jussi
(Ai, Bj, Ci), eg A=Imre B=Jock C=Iago
(Aj, Bi, Ci), eg A=Jan B=Izzy C=Iago
all occur, then (Ai, Bi, Ci ) eg A=Imre B=Izzy C=Iago
also occurs, without which projection followed by reconstitution would create spurious rows.
N-decomposability is clarified by pointing at where it isn't. Consider this cribbed example:
PSC
PID SID Course
~~~~~~~~~~~~~~~~~~~~~~~~
Emile Wendy CSC330
Emile Joan CSC111
Merrie Wendy CSC111
Now projecting PSC to obtain PS(PID,SID), SC(SID,Course), and PC(PID,Course), ie every projection of degree 2, we get:
PS SC PC
PID SID SID Course PID Course
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Emile Wendy Wendy CSC330 Emile CSC330
Emile Joan Joan CSC111 Emile CSC111
Merrie Wendy Wendy CSC111 Merrie CSC111
But neither the projections PS and SC (nor any two of them) are equivalent to PSC: joining them over the common domain SID yields mythical rows (marked *):
PS_SC PS_SC_PC
~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~
PID SID Course PID SID Course
Emile Wendy CSC330 Emile Wendy CSC330
Emile Wendy CSC111* Emile Wendy CSC111*
Emile Joan CSC111 Emile Joan CSC111
Merrie Wendy CSC330* Merrie Wendy CSC111
Merrie Wendy CSC111
Bottom line:
BCNF = every FD is a consequence of the candidate keys
4NF = every MVD is a consequence of the candidate keys
5NF = every JD is a consequence of the candidate keys
Last updated 23 Nov 2021 |