Given FD's:
B->D
D->B
AB->C
Decomposition: {R1(B,D) , R2(A,D,C)}
What fd's will hold on R2??
A -> something
D -> something
C -> something
AD -> something
AC -> something
DC -> something
ADC -> something
Now let's replace "something" by proper values that we get from fd's.
A -> A
D -> DB
C -> C
AD -> ADBC
AC -> AC
DC -> DBC
ADC -> ADCB
Now filter out fd's that include only those attributes that are present in our decomposed relation R2(A,D,C)
A -> A
D -> D
C -> C
AD -> ADC
AC -> AC
DC -> DC
ADC -> ADC
Excluding trivial fd's we get:
AD -> C
Now we have found fd's that hold on R2.
Can we deduce AB - > C from these fd's??
(NO!).
Likewise deduce fd's for R1.
Can we deduce AB - > C from these fd's that hold on R1??
(NO!).