Functional Dependencies provement

Question

Consider the relation R(A,B,C,D,E) and suppose the following functional dependencies:

• AB -> C
• AE -> D
• D -> B

How can I prove that AE is the key?

Example 2

Consider a relation with five attributes A,B,C,D and E and following dependencies:

• AB -> C
• CD -> E
• DE -> B

How can I prove that ABD is a key and AB is not a key?

Example 3

• AB -> C
• AE -> D
• D -> B

Prove that ADE -> BC?

Answer 1

For the first question: AE is a key since AE -> D, i.e., if you know A and E you also know D; since you know D, you also know B (D -> B), and finally, since you know A and B you also know C (AB -> C). Hence, if you know A and E you know all other attributes, hence, AE is a key.

For the second question, you can use the same technique to show that ABD is a key. To show that AB is not a key you apply the same technique and discover that you cannot infer a certain attribute (D).

For the third question, if you know A, D and E then you also know B (D -> B) and since you know A and B you also know C.