Consider a schema with the attribute set ABCDFG and the following FDs:

AB --> CD, BC --> FG,A --> G, G --> B, C --> G.

(a) Find a minimal cover of this set of FDs.
(b) Is the decomposition of the previous schema into ABCD and CFG lossless?

(a) Find a minimal cover of this set of FDs.
Step 1
AB --> C
AB --> D
BC --> F
BC --> G
A --> G
G --> B
C --> G
Step 2 - Reduce the left-hand sides
Since A+ = AGBCD (and so A-->B is entailed), we can replace the first two FDs with
A --> C, A --> D.
Similarly, C+ = CGBFG, so FDs 3 and 4 can be replaced with C --> F, C --> G (the latter is
a duplicate and can be deleted).
Step 3 - eliminate redundant FDs
A --> G is redundant due to A --> C, C --> G. So, we end up with the following minimal
cover:
A --> C
A --> D
C --> F
G --> B
C --> G

(b) Is the decomposition of the previous schema into ABCD and CFG lossless?
The decomposition into ABCD and CFG is lossless, since the intersection is C and C+ =
CFGB. In particular, the FDs imply C -> CFG, which implies losslessness according to the
losslessness criteria.