Equivalence of functional dependencies in DBMS

Finding a relationship between two to FD (functional dependency) sets 

Consider S1 and S2 are two FD sets for a relation R.

1. 1. If all FDs of S1 can be determined from FDs that are present in S2, we can conclude that S2 ⊃ S1.
   2. If all FDs of S2 can be determined from FDs that are present in S1, we can conclude FD1 ⊃ FD2.
   3. If 1 and 2 are satisfied then, S1=S2.

Check for the equivalence of functional dependencies for A relation R (A, B, C, D) having two FD sets S1 = {A->B, B->C, AB->D} and S2= {A->B, B->C, A->C, A->D} 

Solution: 

• Step 1.  Checking whether all FDs of set S1 is present in set S2 
  • FD A->B in set S1 is present in set S2.
  • FD B->C in set S1 is also present in set S2.
  • FD AB->D in present in set S1 but not in S2 but we have to check whether we can derive it or not. In set S2, (AB) + = {A, B, C, D}. It means that A,B,C,D can be determined functionally from AB which implies AB->D holds true for set S2
  • As all FDs in set S1 also hold in set S2, S2 ⊃ S1 is true.

• Step 2.  Checking whether all FDs of S2 are present in S1
  • FD A->B in set S2 is present in set S1.
  • FD B->C in set S2 is also present in set S1.
  • FD A->C is present in S2 but not in S1 but we have to check whether we can derive it or not. In set S1, (A) + = {A, B, C, D}. It means that A,B,C,D can be determined functionally from A which implies A->C holds true for set S1.
  • FD A->D is present in S2 but not in S1 but we have to check whether we can derive it or not. In set S1, (A) + = {A, B, C, D}. It means that A,B,C,D can be determined functionally from A which implies A->D holds true for set S1.
  • As all FDs in set S2 also hold in set S1, S1 ⊃ S2 is true.

• Step 3. We can conclude that S1=S2 as both S2⊃ S1 and S1⊃ S2 are true. Hence these two FD sets are semantically equivalent.