EXERCISES

11.1 Consider the parts relvar P from the suppliers-and-parts database. For simplicity, let’s rename attributes PNO, PNAME, COLOR, WEIGHT, and CITY as A, B, C, D, and E, respectively, and let’s use Heath notation once again. Then the following JDs are all defined with respect to the heading of P:

  1. { AC , ABDE }

  2. { ACD , ABDE }

  3. { AE , ABCD }

  4. { AB , ACD , CE }

  5. { AB , ACD , AE }

  6. { AB , BCD , DE }

  7. { ABC , ACDE , CE }

  8. { ABCD , BDE , BCE }

  9. { AB , ABC , BCD , CDE , AD }

  10. { AB , BC , CD , DE , AD }

  11. { ABD , CDE , ABC , BE , ABE }

  12. { A , AB , ABC , ABD , ACE }

Which of these JDs are trivial? Which ones involve irrelevant components? Which imply which others in the list? Which pairs are equivalent to one another? Which are satisfied by the sample value of relvar P shown in Figure 1-1? Which hold in relvar P? Which are irreducible with respect to P?

11.2 The dependencies in this exercise are all defined with respect to a heading consisting of attributes ABCD.

  1. Does the set of FDs {A → B, A → C} imply the JD {AD,ABC}?

  2. Does the set of FDs {C → D, B → C} imply the JD {AB,BC,CD}?

  3. Does the set of FDs {A → B, B → C} imply the JD {AB,BC,CD}?

  4. Does the the JD {BC,ABD} imply the JD {AB,BC,CD}?

11.3 We know from Exercise 5.4 that the converse of Heath’s Theorem is false. However, there’s an extended version of that theorem whose converse is true. Here it is:

  • Heath’s Theorem (extended version): Let relvar R have heading H and let X, Y, and Z be subsets of H such that the union of X, Y, and Z is equal to H. Let XY denote the union of X and Y, and similarly for XZ. If R is subject to the FD X → Y, then (a) R is subject to the JD {XY,XZ}, and (b) XZ is a superkey for R.

Prove part (b) of this theorem. Prove also that (a) and (b) together imply that X → Y holds (the converse of the extended theorem).

11.4 Consider the following JDs, both of which hold in relvar S:

      { { SNO , SNAME , CITY } , { CITY , STATUS } , { SNAME , CITY } }
      { { SNO , SNAME , CITY } , { CITY , STATUS , SNAME } }

I pointed out in the body of the chapter (in the section COMBINING COMPONENTS) that although the first of these JDs implied the second, decomposing relvar S on the basis of that second JD (even though it’s irreducible) wouldn’t be a good idea. Why not?

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