Problem $1$: Given relation $R(A,B,C,D)$ with FD$\text{'}$s $AB$ and $BC.$

a$.$ Complete the closures for all subsets of R using the FDs provided.

b$.$ Determine all non$-$trivial FDs associated with our relation R and the given FDs$.$

c$.$ Identify the key and all superkeys for relation R$.$

Keys:

Superkevs:

d$.$ Is the set of FDs given minimal? If not, find the minimal basis?