Final answer:
To find the functional dependencies, we look for patterns in the relation instance where the knowledge of one attribute or a combination of attributes consistently determines another. X → Y and XZ → Y are valid functional dependencies because the value of Y is linked and determined by the value of X or the combination of X and Z in the given dataset.
Step-by-step explanation:
To determine the functional dependencies in a set of relation instances, we analyze the values and find dependencies where one set of attributes uniquely determines another. The provided table can be interpreted, and the dependencies found by observing patterns in the values.
When we see the attribute pair (X, Z) has the same value of Y associated with it consistently, it indicates a functional dependency. In the supplied data, whenever X is x1, Y is always y1, regardless of Z's value. This suggests a functional dependency of X → Y. Similarly, a consistent pairing of Z and Y would hint at Z → Y, but here we cannot claim it as Z appears with different Y values. Furthermore, when we combine X and Z together, we find that a particular combination of X and Z always corresponds to the same Y, leading to the functional dependency XZ → Y.
To summarize the thought process further: A functional dependency X → Y means that if we know X, we can determine Y. However, knowing Y does not necessarily allow us to determine X. We are looking for such relationships in the relation instance. The dependencies X → Y and XZ → Y are evident because the value of Y is consistent whenever X or the combination of X and Z is the same. However, Z does not fulfill this condition by itself, therefore Z → Y is not a valid functional dependency based on the given data.