To find the canonical cover of the given set of functional dependencies, we need to first simplify the given set of functional dependencies by removing redundant dependencies and then find the minimal cover. The steps to find the canonical cover are as follows:
Step 1: Remove redundant dependencies
We can see that the dependencies CF->BD and ACD->B are redundant because they can be inferred from the other dependencies. Hence, we can remove them.
The simplified set of functional dependencies is:
f1: AB -> C
f2: C -> A
f3: BC -> D
f4: BE -> C
f5: CE -> F
f6: D -> E
Step 2: Find the closure of each functional dependency
f1: AB+ = ABCDEF
f2: C+ = CADEFB
f3: BC+ = BCDEF
f4: BE+ = BECFAD
f5: CE+ = CEFAD
f6: D+ = DE
Step 3: Remove redundant dependencies using the Armstrong's axioms
We can use the Armstrong's axioms to check for redundant dependencies. The axioms are:
Reflexivity: If Y is a subset of X, then X -> Y.
Augmentation: If X -> Y, then XZ -> YZ for any Z.
Transitivity: If X -> Y and Y -> Z, then X -> Z.
Using these axioms, we can check for redundancy as follows:
f1: AB+ = ABCDEF
f1+: AB -> CDEF
f2: C+ = CADEFB
f2+: C -> ADEFB
f3: BC+ = BCDEF
f3+: BC -> DEF
f4: BE+ = BECFAD
f4+: BE -> CFAD
f5: CE+ = CEFAD
f5+: CE -> FAD
f6: D+ = DE
f6+: D -> E
Step 4: Find the minimal cover
We can remove redundant dependencies from the simplified set of functional dependencies using the Armstrong's axioms. The minimal cover is:
f1: AB -> C
f2: C -> A
f3: BC -> D
f4: BE -> C
f5: CE -> F
f6: D -> E
Therefore, the canonical cover of the given set of functional dependencies is:
AB -> C
C -> A
BC -> D
BE -> C
CE -> F
D -> E
