Let R$($ABCD$)$ be a relation with functional dependencies $\{$A $->$ B$,$ C $->$ D$,$ AD $->$ C$,$ BC $->$ A$\}.$ Which of the following is a lossless$-$join decomposition of R into Boyce$-$Codd Normal Form $($BCNF$)?$

Note: The "lossless$-$join" property is the generalization of the property discussed in Sect. $3\mathrm{.}6\mathrm{.}5($p$.112),$ where a test is given to show that the decomposition of a relation into two yields no more tuples when we project onto the relations and then rejoin. The extension to any number of relations is the same: take the projection of the given relation onto each of the relations of the decomposition, and then rejoin $($in any order $---$ it doesn't matter because natural join is commutative and associative$).$ If the resulting join has no more tuples than the original, the join is lossless. In this question, you are really asked to generalize the idea of Sect. $3\mathrm{.}6\mathrm{.}5.$ If you have trouble, the choice explanations will give you a hint.

a$)\{$AB$,$ AC$,$ AD$,$ BC$,$ BD$,$ CD$\}$

b$)\{$AB$,$ AC$,$ BCD$\}$

c$)\{$AB$,$ AC$,$ AD$,$ BCD$\}$

d$)\{$AC$,$ BD$,$ CD$\}$