Given the set F= {A-B, AB-C, AC—BD} of functional dependencies, prove the following dependencies by using the Armstrong axioms. 

(1) A-ABC (2) AD-BCD 

2. Consider a relation schema R(X, Y, Z) with the functional dependencies XYZ and Z-X. Can we conclude that Y-XZ holds? If yes, please argue why. If no, please argue why not by giving a counterexample. 

3.Consider the relation schema R(A, B, C, D, E, F) and the set of functional dependencies F = {A-B, A-C, CD-E, CD—F, B-E}. Infer at least five new FDs by using five different Armstrong's axioms and derived inference rules. (Please do not include the trivial ones such as A-A in your answer.) Show each step. 

4. Assume we have a set F = {A-B, C-D} of functional dependencies for a relation schema R(A, B, C, D). Write down all the functional dependencies of the closure F of F and count them.