Advanced Database Management System - Tutorials and Notes: Closure of Set of Functional Dependencies - Example

## Finding Closure of Set of Functional Dependencies Examples / Solved Examples - Closure of Set F of Functional Dependencies / Example for finding F+ / What is meant by the closure of a set of functional dependencies illustrate with an example

Closure of Set F of Functional Dependencies can be found from the given set of functional dependencies by applying the Armstrong's axioms. Closure is denoted as F+.

Recall the axioms;

 Reflexivity rule If X is a set of attributes, and Y is subset of X, then we would say, X → Y. Augmentation rule If X → Y, and Z is another set of attributes, then we write XZ → XY. Transitivity rule If X → Y, and Y → Z, then X → Z is true. Union rule If X → Y and X → Z, then X → YZ is true. Decomposition rule Its reverse of Rule 4. If X → YZ, then X → Y, and X → Z are true Pseudotransitivity rule If X → Y and ZY → A are true, then XZ → A is also true.

### Example 1 - Finding F+ of given set F of functional dependencies:

Consider a relation R with the relation schema R = (A, B, C, D, E). Assume that the relation holds the following set of functional dependencies. That is, F is given as follows;
F = { A → BC, CD → E, B → D, E → A}
Find the F+ (Closure).

From A → BC and B → D, we can deduce A → D (By applying Transitivity rule)
Through A → BC, A → D, and CD → E, we can derive A → E (By applying Transitivity rule)
Through above deductions, we can write A → ABCDE.

From E → A, and A → ABCDE, we can derive E → ABCDE
From CD → E and E → ABCDE, then CD → ABCDE
From B → D, we can derive BC → CD using Augmentaion rule. It would further mean, BC → ABCDE

Like this, we can continue to derive all the possible functional dependencies.

We would write all the functional dependencies as follows to get F+;

F+ = { A → BC, CD → E, B → D, E → A, A → D, A → E, A → ABCDE, BC → CD, CD → ABCDE }

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