A completely non-trivial relationship occurs when there’s no intersection between the attributes in P and Q.
Let’s analyze each option:
- P→Q, Q→R: P is a proper subset of Q, while Q is a proper subset of R. This establishes a completely non-trivial relationship because each determinant determines its dependent attribute(s) independently and without redundancy.
- P→Q, Q→S, R→S: P is a proper subset of Q, which is a proper subset of S. Similarly, R is a proper subset of S. This also guarantees a completely non-trivial relationship.
- P→Q, Q→R, R→S: P is a proper subset of Q, R, and S. This follows the same pattern as the previous options and guarantees a non-trivial relationship.
- P→Q, Q→RS: In this example, Q is not a suitable subset of RS, indicating that the relationship is not entirely non-trivial. The dependency Q→RS means that Q uniquely determines both R and S, but the dependent side has unnecessary properties, making it non-completely non-trivial. For a completely non-trivial relationship, we ideally want dependencies where each determinant provides unique information about its dependent attribute(s) without any overlap or redundancy. In the given dependency, the presence of RS as a composite attribute in the dependent side of the functional dependency violates this principle.