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What problem do you facing even after seeing the solution, do tell ??
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I think the given solution is self-explanatory

first, check if it is in 2NF or not, by checking partial dependencies
then check if the resulting decomposition is in 3NF or not, by checking transitive dependencies
and then go for BCNF

Let us know you are facing any problem.
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for finding BCNF why they have used minimal cover
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They are using minimal cover to “identify the attributes and FD’s which should be included in the new table after decomposition” and along with that they are checking that the dependency can be preserved or not.

And all the above things can be done easily by getting the minimal cover.

That’s the only reason why they are using minimal cover in the solution

They are not using minimal cover to find BCNF.

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