7.14

Show that there can be more than one canonical cover for a given set of functional dependencies, using the following set of dependencies:

$$X\to YZ,Y\to XZ,\text{ and }Z\to XY$$

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Consider the first functional dependency. We can verify that $Z$ is extraneous in $X\to YZ$ and delete it. Subsequently, we can similarly check that $X$ is extraneous in $Y\to XZ$ and delete it, and that $Y$ is extraneous in $Z\to XY$ and delete it, resulting in a canonical cover $X\to Y,Y\to Z,Z\to X$.

However, we can also verify that $Y$ is extraneous in $X\to YZ$ and delete it. Subsequently, we can similarly check that $Z$ is extraneous in $Y\to XZ$ and delete it, and that $X$ is extraneous in $Z\to XY$ and delete it, resulting in a canonical cover $X\to Z,Y\to X,Z\to Y$.