. What is the closure of the attribute set? Support the answer by describing the algorithm and suitable example. Mention the applications of the algorithm.

The closure of an attribute set refers to the set of attributes that can be functionally determined by the given attribute set in the context of a particular set of functional dependencies.

The algorithm to find the closure of an attribute set is as follows:

1. Start with the closure as the set of attributes itself.
2. For each functional dependency, if the determinant (left-hand side) is a subset of the closure, add the dependent (right-hand side) to the closure.
3. Repeat step 2 until the closure set stops changing.

Let's illustrate this with an example:

Consider a relation R(A, B, C, D, E) with the following functional dependencies:

1. AB -> C
2. C -> D
3. D -> E

We want to find the closure of the attribute set {A, B}.

1. Start with the closure as {A, B}.
2. From the first functional dependency, AB -> C, since AB is a subset of the closure, add C to the closure. The closure is now {A, B, C}.
3. From the second functional dependency, C -> D, since C is a subset of the closure, add D to the closure. The closure is now {A, B, C, D}.
4. From the third functional dependency, D -> E, since D is a subset of the closure, add E to the closure. The closure is now {A, B, C, D, E}.
5. No more changes can be made to the closure, so the algorithm stops.

So, the closure of the attribute set {A, B} is {A, B, C, D, E}.

The closure of an attribute set is used in various applications:

1. Testing for Superkey: If the closure of an attribute set includes all attributes of the relation, then the attribute set is a superkey.

2. Testing for Candidate Key: If the closure of an attribute set includes all attributes of the relation and there is no proper subset of the attribute set whose closure includes all attributes, then the attribute set is a candidate key.

3. Testing for Functional Dependency: A functional dependency X -> Y holds if and only if Y is a subset of the closure of X.

4. Normalization: The closure of an attribute set is used in the process of normalization, particularly in the decomposition of relations to achieve various normal forms.