Closure of an attribute set

1. Closure of an attribute set: The set of those attributes which can be functionally determine from an attribute set is called as a Closure of an attribute set.
2. Closure of attribute set {X} is denoted as {X} + .

Steps to find  Closure of an attribute set

 1. Add the attributes contained in t he attribute set for which closure is being calculated to the result set.

2. Recursively add the attributes to the result set which can be functionally determined from the attributes already contained in the result set.

Example-1:

Consider a relation R ( A , B , C , D , E , F , G ) with the functional dependencies-
A → BC
BC → DE
D → F
CF → G

Now, let us find closure set of some attribute set

Closure of attribute A-

A + = { A }
= { A , B , C } ( Using A → BC )
= { A , B , C , D , E } ( Using BC → DE )
= { A , B , C , D , E , F } ( Using D → F )
= { A , B , C , D , E , F , G } ( Using CF → G )
Thus,
A+ = { A , B , C , D , E , F , G }

Closure of attribute D-

D+ = { D }
= { D , F } ( Using D → F )
We can not determine any other attribute using attributes D and F contained in the result set.
Thus,
D+ = { D , F }

 Closure of attribute set {B, C}-

{ B , C } + = { B , C }
= { B , C , D , E } ( Using BC → DE )
= { B , C , D , E , F } ( Using D → F )
= { B , C , D , E , F , G } ( Using CF → G )
Thus,
{ B , C } + = { B , C , D , E , F , G}