How to find attribute closure using Functional Dependency

For a Given set of Functional dependencies, the attribute closure of an attribute A will be a set S determined as below:

1. Add A to the set S.
2. Add all the attributes which are derived from A to the set S
3. Add all the attributes which are functionally determined by the attributes of set S, recursively.

Attribute closure is denoted by A⁺

Example:

Given a Relation R(A, B, C, D, E) {AB -> CD, D -> E, A -> C,B -> D}.

we can determine A⁺ as:    

1. Add A to the set S = {A}.
2. get all the attribute which are derived from A
   • A->C (From given Functional Dependencies)
   • Add C to the set. Now S will contain {A,C}
3. No other attribute can be derived from A and C hence

A⁺= {A,C}.

we can determine B⁺ as:

1. Add B to the set S = {B}.
2. get all the attribute which are derived from B
   • B->D (From given Functional Dependencies)
   • Add D to the set. Now S will contain {B,D}
3. get all the attribute which are either derived by D or the combination of BD
   • D->E
   • Add E to the Set. Now S will contain {B,D,E}
4. No other attribute can be derived from B,D,E or with their combination hence

B⁺= {B,D,E}.

Similarly we can get closure of any attribute.