Consider a relation schema with attributes ABCGWXYZ and the set of dependencies F = {XZ ? ZYB, YA ? CG, C ? W , B ? G, XZ ? G}. Solve the following problems using the appropriate algorithms.

Question

a. Find a minimal cover for F.
b. Is the dependency XZA ? YB implied by F?
c. Is the decomposition into XZYAB and YABCGW lossless?
d. Is the above decomposition dependency preserving?

Answer 1

a. Minimal cover: After splitting the right-hand sides, we get XZ ? G plus the following
set:
XZ ? Y
XZ ? B
YA ? C
YA ? G
C ? W
B ? G

It is easy to see that the left-hand sides XZ and YA (as well as C and B) cannot be reduced because there are no FDs of the form X ? ..., Y ? ...,Z ? ...,A ? ..., or {}?....
Then, we notice that XZ G is redundant and none of the other FDs is, so the minimal cover consists of the above siz FDs.

b. Yes, because XZA+= XZAYBCWG, which includes YB.
c. Yes, because XZYAB ? YABCGW = YAB and YAB+= YAMCGW .
d. Yes, because every FD in the above minimal cover is embedded into either XZYAB or YABCGW .