Consider the following set F of functional dependencies on the relation schema r(A,B,C,D,E,F):
I have tried to reach out to my instructor with no luck and I really want to understand this process, but no matter how much I read the material I cannot seem to make this fit in my little brain. Can someone please help me with the following questions?
A-->BCD
BC-->DE
B-->D
D-->A
a. Compute B+.
I believe that this one is as follows. Does this seem to be correct?
B+ denotes closure of B.
B --> D
B+ = {BD}
D --> A
B+ = {ABD}
A --> BCD
B+ = {ABCD}
BC --> DE
B+ = {ABCDE}
All the attributes of the relation can be found by B. So, B is the primary key of the relation.
b. Prove (using Armstrong’s axioms) that AF is a superkey.
I do not understand what to do with F, because it does not show up in the above relationships.
c. Compute a canonical cover for the above set of functional dependencies F; give each step of your derivation with an explanation.
d. Give a 3NF decomposition of r based on the canonical cover.
Answer
All the attributes of the relation can be found by B. So, B is the primary key of the relation.
No. If B could determine all the attributes of the relation, B would be a candidate key. There might be more than one candidate key, and there's no formal reason to identify one candidate key as "primary" and others as "secondary".
But B doesn't determine all the attributes of the relation. It doesn't determine F.
I do not understand what to do with F, because it does not show up in the above relationships.
Speaking informally, if an attribute doesn't show up on the right-hand side of any functional dependencies, it has to be a part of every superkey.
r = {ABCDEF}
To prove that AF is a superkey (or candidate key), compute the closure of AF for the relation R = {ABCDEF}. Use the same FDs above.