Consider a schema R and a set F of functional dependencies as follows: R = (A,B,C,D,E). F = {A → BC,CD → E,B →D,E → A}. Determine whether or not the following decomposition of R is a lossless decomposition: R = (A, B, C), R2 = (A, D, E). Show your computational steps as you derive the answer.

4. Draw the Hasse Diagram for the following partial orderings.(a) S = {a, b, c} and ρ = {(a, a),(b, b),(c, c),(a, b),(b, c),(a, c)}(b) S = {a, b, c, d} and ρ = {(a, a),(b, b),(c, c),(d, d),(a, b),(a, c)}(c) S = {∅, {a}, {a, b}, {c}, {a, c}, {b}} and AρB ↔ A ⊆ B.

Consider the relation R on Z? given by: R = {((a, b), (c, d)) : a +d = c+b} Prove that R is an equivalence relation, and determine 3 elements of [(1,3)], the equivalence class of (1, 3).

Consider a schema R, a set F of functional dependencies on R, and two candidate keys (k1, k2) as follows R= (A,B,C). F = {B →C,AC –→ B}. ki = AC. k2 =AB. (a) Is R in BCNF?

The composition of Figure 2 has 6 states — how many many traces does it have?

Dissect NP and P, then provide concrete instances to show how they vary in use.

Consider an equivalence relation R on X={a,b,c}. R= {(a,a), (a,c), (b,b), (c,a), (c,c) } [a]=

Consider a relation R(A, B, C, D, E, F), with FDs AB → C , BC → D, D → E, BE → F and DF → C.1. Is R in BCNF? Explain why or why not.2. Find the closure of the following:1. (A, B)2. (B, D)3. (D, F)3. List one superkey of R that is not a (minimal) key

show the steps used to asnwer this question what are the star values?

1.1Although the BCNF algorithm ensures that the resulting decomposition is lossless,it is possible to have a schema and a decomposition that was not generated by the algorithm, thatis in BCNF, and is not lossless. Give an example of such a schema and its decomposition. 1.2Show that every schema consisting of exactly two attributes must be in BCNFregardless of the given set F of functional dependencies.