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CS 377 - Database Systems
Function Dependency & Decomposition Homework

Due: See class webpage

(Question 14.26 in text book)
Conside the relation
```
   R = (A, B, C, D, E, F, G, H, I, J)
```
and the following set of functional dependencies on R:
```
   F = { A, B -> C
         A -> D, E
	 B -> F
	 F -> G, H
	 D -> I, J
       }
```
- What is the key(s) in R
- Decompose R losslessly into BCNF
(Question 14.27 in text book)
Conside the relation
```
   R = (A, B, C, D, E, F, G, H, I, J)
```
and the following set of functional dependencies on R:
```
   F = { A, B -> C
         B, D -> E, F
	 A, D -> G, H
	 A -> I
	 H -> J
       }
```
- What are the key(s) in R
- Decompose R losslessly into BCNF

(Question 14.28 in text book)

Consider the following relation:

    Tuple#       A         B          C
    ----------------------------------------
      #1         10        b1         c1
      #2         10        b2         c2
      #3         11        b4         c1
      #4         12        b3         c4
      #5         13        b1         c1
      #6         14        b3         c4

Given the above database content, which of the following functional dependencies may hold in the above relation ? If the functional dependency cannot hold, explain why by specifying the tuples that cause the violation.
```
   1. A -> B
   2. B -> C
   3. C -> B
   4. B -> A
   5. C -> A
```
Does the above relation have a potential candidate key ? If it does, what is it ? If it does not, why not ?

(Adpated from Question 14.29 in text book)
Consider a relation:
```
   R = (A, B, C, D, E)
```
with the following functional dependencies:
```
   A, B -> C
   C, D -> E
   D, E -> B
```
- Compute the closure AB⁺
- Compute the closure CD⁺
- Compute the closure DE⁺
(Adpated from Question 14.29 in text book)
Consider a relation:
```
   R = (A, B, C, D, E)
```
with the following functional dependencies:
```
   A, B -> C
   C, D -> E
   D, E -> B
```
- Find all the keys in relation R