Need assistance in Boyce Codd Normal Form

Question

I am stuck up in Normalization. I have normalized till 3NF. However I do not how to perform BCNF. The 3NF table list are shown below.

1. Table1 ( A , B, C); FD: A -> B, A -> C.
2. Table2 ( D , E); FD: D -> E.
3. Table3 ( A , D , F); FD {A, D} -> F
4. Table4 ( A , D , H , I , J, K); FD: {A, D, H, I} -> J.
   
   As mentioned in the above example, I do not know how to apply BCNF for the Table4. Attribute K does not have any dependency as per the requirements. I assume that we have to apply BCNF in Table4. Requesting you to please help with this regard. Thank you

Answer 1

Armstrong's Axiom:

Axiom of augmentation : If X->Y, then XZ->YZ for any Z

So,this can be applied to table 4,

IN Table 4,FD: {A, D, H, I} -> J. can be written as FD: {A, D, H, I,K} -> {J,K}

Then,we can apply decomposition axiom,

Decomposition : If X -> YZ then X -> Y and X -> Z

So,applying to table4,we get,

FD : {A, D, H, I,K} -> J,
FD : {A, D, H, I,K} -> K

so,the two FDs of table4 are:

FD : {A, D, H, I} -> J,
FD : {A, D, H, I,K} -> {J,K}

Thus {A,D,H,I,K} is a Super key ,by definition of BCNF ,we get table4 is in BCNF

Hope this helps.