Question $7.$ Consider the following set of functional dependencies:

$=\{CFAD,CAB,FDDE,CA,EC,DBF,EA\}$

$($a$)$ Provide a proof of $|==CDEF,$ as a numbered sequence of functional

dependencies entailed by $,$ terminating in $CDEF.$ Each step in the

sequence must be explicitly justified by either indicating that the functional

dependency is from $,$ or stating which axiom was used to derive it$,$ along

with references to the functional dependencies to which the axiom was ap$-$

plied.

Four marks are awarded for a correct proof that may use all essential Arm$-$

strong's axioms $($reflexivity$,$ augmentation, transitivity$)$ as well as decom$-$

position and union; two extra marks are awarded if$,$ in addition, the proof

does not use decomposition nor union, and its length does not exceed $13.$

$($b$)$ Find a minimal cover of $.$ Justify your answer by providing details of each

step, including the working of all the entailments one needs to check.