1. Third Normal Form

2. BCNF and Dependency Preservation
theater
play
Wharton Center
Nutcracker
Children's Theater
R = {play, theater, city}
FDs = {
theater → city (each theater is in one city)
play city → theater (only one theater is allowed to hold the play in each city) }
Keys:
{play, city}
{theater, play}
BCNF Decomposition:
{theater, play} and {theater, city}
Does the following data obey the functional dependencies?
No! But you can only know with a join.
theater
city
Wharton Center
East Lansing
Children's Theater

3. Third Normal Form
Third normal form is more permissive than BCNF, and allows dependency preservation.
A relation R is in third normal form (3NF) if:
Whenever there is a nontrivial FD A1, A2, ..., An → B1, B2, ... Bn for R,
Either:
It is the case that {A1, A2, ..., An} is a superkey for R.
or, all of the B's that aren't in A are a member of some key.
The bolded part is what 3NF allows that BCNF doesn't.

4. 3NF Decomposition
Input: A relation R with a set of functional dependencies F.
Output: A decomposition of R0 into a collection of relations, each of which is in 3NF. The decomposition has the lossless-join and dependency-preservation properties.
Method: Perform the following steps
Find a minimal basis for F, say G (the minimal basis are the FD's that don't imply each other)
For each functional dependency (X → A) in G, use XA as the schema of one of the relations in the decomposition.
If none of the relation schema from Step 2 is a superkey for R, add another relation whose schema is the key for R.

5. R = {A, B, C, D, E}
FDs = {AB → C; C → B; A → D; AB → D}
Step 1: Find minimal basis
AB → D is implied by (A → D) so it is ignored
Minimal basis is {AB → C; C → B; A → D}
Step 2: Make schema from each FD
(AB → C) becomes {ABC}
(C → B) becomes {BC} # is a subset of other relations so can be ignored
(A → D) becomes {AD}
Result = {ABC} and {AD}
Step 3: Make sure there's a superkey relation
Neither {ABC} or {AD} is a superkey
So add {ABE} (which is a key)
Final result = {ABC} and {AD} and {ABE}

6. AnimeTraits ( name, hair_color, tsundere, eye_size) FDs =
Is decomposition into Hair(name, hair_color, eye_size) and Emotion(hair_color, tsundere) a valid (lossless join and dependency preserving) 3NF decomposition?
1. Yes
2. No, it isn't dependency preserving
3. No, the intersection isn't a superkey for a subrelation
4. No, the union of attributes isn't the original relation