Relational Model Practice Problems

1. Which approach to converting an E/R design involving entity sets which participate in isa relationships to a relational design requires more relations: the E/R approach or the OO approach?
   
   
2. Exercise 3.1.2 (p. 65) of [FCDB].
   
   
3. Exercises for §3.2 (pp. 75-76) of [FCDB].
   
   
4. Exercises for §3.3 (pp. 80-82) of [FCDB].
   
   
5. Exercises for §3.4 (pp. 88-89) of [FCDB].
   
   
6. Does the definition of FD preclude the case in which the lhs is the empty set (i.e., {} → A)? If not, explain the meaning of such a dependency?
   
   
7. Prove that Armstrong's transitivity axiom is sound (i.e., every relation which satisfies the FD X → Y and Y → Z must also satisfy the FD X → Z).
   
   
8. Using only Armstrong's axioms and the FD's: {AB → C, A → BE, C → D}, give a complete derivation of the FD A → D.
   
   
9. Exercises for §3.5 (pp. 100-102) of [FCDB].
   
   
10. (true/false) Does the following relation instance satisfy the A → B and BC → A FD's?

    A B C
    -----
    1 2 3
    2 2 2
    1 3 2
    4 2 3
    

    
    
11. Prove or disprove the following inference rules for FD's. A proof can be made either by a proof argument or by using Armstrong's axioms. A disproof should be performed by demonstrating a relation instance which satisfies the conditions and FD's on the lhs of the inference rule, but does not satisfy the FD's on the rhs.
    
    
    1. {W → Y, X → Z} |= {WX → Y}
    2. {X → Y} and Z ⊆ Y |= {X → Z}
    3. {X → Y, X → W, WY → Z} |= {X → Z}
    4. {XY → Z, Y → W} |= {XW → Z}
    5. {X → Z, Y → Z} |= {X → Y}
    6. {X → Y, XY → Z} |= {X → Z}
    7. {X → Y, Z → W} |= {XZ → YW}
    8. {XY → Z, Z → X} |= {Z → Y}
    9. {X → Y, Y → Z} |= {X → YZ}
    10. {XY → Z, Z → W} |= {X → W}
    
    
12. Are the following two sets of FD's F and G equivalent?
    
    F = {A → C, AC → D, E → AD, E → H}
    G = {A → CD, E → AH}
    
    
13. Consider the following sets of FD's over the relation R(A, B, C):
    • F₁ = {A → B, B → C}
    • F₂ = {A → B, A → C}
    • F₃ = {A → B, A B → C}
    
    Identify all sets above which follow from a) F₁ b) F₂, and c) F₃. Which pairs of these sets of FD's (F₁ and F₂, F₂ and F₃, and F₁ and F₃) are equivalent to each other?
    
    
14. Consider the following set of FD's over the relation Widgets(source, destination, volume, weight):
    
    source volume -> weight
    destination -> weight
    source volume -> destination
    
    
    1. Give a minimal set of FD's equivalent to the set above.
    2. Give one key for the relation Widgets and prove it is a key.
    3. Any there any other keys? Give reasons.
    
    
    
15. Exercises for §3.6 (p. 117) of [FCDB].
    
    
16. Exercises for §3.7 (pp. 126-127) of [FCDB].