Database Systems Question: Please answer ALL parts of the questions with FULL explanations. Thx

Problem 2: Checking Functional Dependencies

For each part of this problem you will provide a single SQL query to check whether a certain condition holds on a specific instance of a relation in the following way: your query should return an empty result if and only if the condition holds on the instance. (If the condition doesn't hold, your query should return something non-empty, but it doesn't matter what this is).

Note our language here: the conditions that we specify cannot be proved to hold in general without knowing the externally-defined functional dependencies. This means that you are checking whether they could hold in general for the relation, given any specific set of tuples.

You may assume that there will be no NULL values in the tables, and you may assume that the relations are sets rather than multisets, but otherwise your query should work for general instances. We define the schemas of the tables used below for convenience, but in this problem you will need to construct your own test tables if you wish to use them to check your answers!

a){A,B} -> {C} and {C} -> {A,B} holds for relation R(A,B,C,D,E)

b){A,B,C} is a superkey for a relation R(A,B,C,D,E)

c){A} and {C} are keys for a relation S(A,B,C)