How To Find Minimal Set?

Pre-requisite-Attribute and keys in DBMS and Types of functional dependency

If we have a set of functional dependencies, we get the simplest and irreducible form of functional dependencies after reducing these functional dependencies. This is called the Minimal Cover or Irreducible Set (as we can’t reduce the set further). It is also called a Canonical Cover.

Let us understand the procedure to find the minimal cover by this example:

The Given Functional Dependencies are – A->B, B ->C, D->ABC, AC-> D

We can find the minimal cover by following the 3 simple steps.

Step: 1   First split the all left-hand attributes of all FDs (functional dependencies).

A->B, B->C, D->A, D->B, D->C, AC->D
[Note: We can't split AC->D as A->D, C->D]

Step: 2 Now remove all redundant FDs.

[Redundant FD is if we derive one FD from another FD ]
Let, 's test the redundance of A->B 
A+ = A (A is only closure contains to A,
 simply we can derive A from A)
So, A->B is not redundant.
Similarly, B->C  is not redundant.
 But, D->B and D->C is redundant 
because D+= A and A+=B, So D+=B can be
derived which means D->B is redundant.
So, We remove D->B from 
the FDs set.
Now, check for  D->C, it is not redundant.
because we can't D+=B and B+=C as we remove D->B from the list.
At last, we check for AC->D. This is also not redundant.
AC+=AC 
So, the final FDs are: A->B, B->C, D->A,D->C, AC->D

Step: 3 Find the Extraneous attribute and remove it.

In this case, we should only check ->D. Simply 
we can say the right-hand attributes are pointed
by only one attribute at one time.          
AC->D, either A or C, or none can be extraneous.
If A=+ C then C is extraneous and it can be removed.
If C+=A then A is extraneous and it can be removed.
So, the final FDs are: A->B, B->C, D->A,D->C, AC->D

Hence, we can write it as A->B, B->C, D->AC, AC->D this is the minimum cover.