Any help with these questinos is much more appriciated

(1) Use Armstrong's axioms and rules to prove that

F = {A -> B, BC->DE, AD->G}

implies AC -> G

(2) Consider R(A,B,C,D,E) with

F = {AC->B, CD->B, D->E, A->C}

(a) What are A+, B+, C+, D+ and E+?

(b) What are the candidate keys? Why?

(c) Show all prime attributes and non-prime attributes?

(d) Give a canonical cover of F?

(e) What is the highest normal form (up to BCNF) of R? Why?

(f) If R is not in BCNF, can you provide a lossless FD preserving decompositions of R into BCNF relations?

(3) Consider R(A,B,C,D,E,F) with

F= {A->D, CE->BF, AF->D, BD->C}

(a) What are A+, B+, C+, D+, E+ and F+?

(b) What are the candidate keys? Why?

(c) Show all prime attributes and non-prime attributes?

(d) Give a canonical cover of F?

(e) What is the highest normal form (up to BCNF) of R? Why?

(f) If R is not in BCNF, can you provide a lossless FD preserving decompositions of R into BCNF relations?

(4) It is known that a relation R with arity 3 is in 3NF and has no composite candidate key. Prove that it is also in BCNF.

(5) Consider that R(A,B,C,D) with a canonical cover of {A->B, X->A} where X is a subset of {C,D}. List all candiate keys. Analyze the highest normal form for various possibility of X (i.e., C, D or CD).

(6) Consider R(A,B,C,D,E) with {BC->A, D->AE, B->C}

It is decomposed into R1(C,D), R2(B,D) and R3(A,D,E).

Is the decomposition lossy? You must use the chase matrix algorithm (EN Algorithm 16.3) to show your reasoning.