How Do Derived Functional Dependencies Impact Key Candidate Identification?

[mathmari]

Hey!

Given the relational scheme $R = (A, B, C, D)$ with the set of functional dependencies $F =\{AB \rightarrow C, \ BC \rightarrow D, \ CD \rightarrow A, \ AD \rightarrow B\}$.

1. Give all non-trivial dependencies that can be derived from F. Justify your derived dependencies.
2. What are the key candidates?Could you explain to me what we have to do at 1 ? :unsure:

 

[Valkarie]

1. Non-trivial dependencies that can be derived from F include:A $\rightarrow$ D, B $\rightarrow$ D, A $\rightarrow$ C, and B $\rightarrow$ C. To justify these dependencies, we can use Armstrong's Axioms. For example, for the dependency A $\rightarrow$ D, we can use the transitivity axiom to derive it since we have AB $\rightarrow$ C and BC $\rightarrow$ D. Since C is a subset of both AB and BC, we can conclude that A $\rightarrow$ D. 2. The key candidates for this relational scheme are AB and CD. These combinations of attributes have all the other attributes in their respective closure sets and therefore form keys.