1. Mulitvalued Dependencies

2. What functional dependencies are obeyed? 1. name → area_code phone
birthdate
Josh
253
517
Hancheng
382
Tyler
What functional dependencies are obeyed?
1. name → area_code phone
2. name →
3. name → birthdate
4. None of the above
Note, the data is made up, don't get any presents.

3. name →→ area_code phone name →→ email
Josh
253
517
Hancheng
382
Tyler
name
birthdate
Josh
Hancheng
Tyler
But there is still redundancy!!!
There's multiple combinations of phone numbers and .
name →→ area_code phone
name →→
name
Josh
Hancheng
Tyler
name
area_code
phone
Josh
253
517
Hancheng
382
Tyler

4. Multivalued Dependency
A multivalued dependency (MVD) is a statement about some relationship R that when you fix the values for one set of attributes, then the values in certain other attributes are independent of that values of all other attributes in the relation.
A1, A2, ..., An →→ B1, B2, ..., Bm
holds for a relation R if when we restrict ourselves to the tuples of R that have particular values for each of the attributes among the A's, then the set of values we find among the B's is independent of the set of values we find among the attributes of R that are not among the A's or B's.
This MVD holds if:
For each pair of tuples t and u of a relation R that agree on all the A's, we can find in R some tuple v that agrees:
With both t and u on the A's
With t on the B's
With u on all attributes of R that are not among the A's or B's.

5. X →→ Y X Y
rest X1 Y1 R1 Y2 R2
...

6. Rules regarding MVD's Trivial MVD's: Transitive Rule: FD Promotion:
A1, A2, ..., An →→ B1, B2, ..., Bm holds in any relation if {B1, B2, ..., Bm } is a subset of {A1, A2, ..., An }
Transitive Rule:
if A1, A2, ..., An →→ B1, B2, ..., Bm and B1, B2, ..., Bm →→ C1, C2, ..., Ck then A1, A2, ..., An →→ C1, C2, ..., Ck
FD Promotion:
Every FD is an MVD
if A1, A2, ..., An → B1, B2, ..., Bm, then A1, A2, ..., An →→ B1, B2, ..., Bm
Complementation Rule:
if A1, A2, ..., An →→ B1, B2, ..., Bm, then A1, A2, ..., An →→ C1, C2, ..., Ck where C are all attributes not among A's and B's

7. No Splitting Rule for MVD
name →→ area_code phone
Cannot be split into:
name →→ phone
name →→ area_code
Why? Because the area_code and phone are a group that together form a unit. If you broke them apart, You would make area_code and phone independent, and all possible combinations would need to be present.

8. Proving All FD's are MVD's
X → Y then X →→ Y X Y
rest X1 Y1 R1
Y2 == Y1 R2
...

9. Fourth Normal Form
This form avoids redundancy regarding multivalued dependencies, and is basically identical in approach to Third Normal Form.
A relation R is in fourth normal form (4NF) if whenever
A1, A2, ..., An →→ B1, B2, ..., Bm
is a nontrivial MVD, and {A1, A2, ..., An} is a superkey.

10. Decomposition into 4NF
Input: A relation R0 with a set of functional and multivalued dependencies S0.
Output: A decomposition of R0 into relations, all of which are in 4NF. The decomposition has the lossless-join property.
Method: Do the following steps, with R = R0:
Find a 4NF violation in R, say A1, A2, ..., An →→ B1, B2, ..., Bm, where {A1, A2, ..., An} is not a superkey. Note this MVD could be a true MVD, or it could be a FD (A1, A2, ..., An → B1, B2, ..., Bm,), since every FD is an MVD. If there is none, return R.
If there is such a 4NF violation, break the schema for the relation R that has the 4NF violation into two schema:
R1, whose schema is A's and B's.
R2, whose schema is the A's and all attributes of R that are not among A's and B's.
Find the FD's and MVD's that hold in R1 and R2. Recursively decompose R1 and R2 with respect to the respective dependences.