six attributes A, B, C, unctional CR→DADFC-
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- Show that the following decomposition of the schema R of Exercise 7.1 is not alossless decomposition:(A, B, C)(C, D, E).Hint: Give an example of a relation r(R) such that ΠA, B, C (r) ⋈ ΠC, D, E (r) ≠ rConsider a relation schema R(A, B, C, D, E, G) and a set of functional dependencies F = {A → C, AD → CE, B → ACD, C → B}. (a) Show the steps of computing a canonical cover for F. (b) Compute the closure of AG and then determine whether or not AG is a candidate key. (c) Determine whether or not (A, E, G) is in BCNF and justify your answer using the transitive closure of a set of attributes. If (A, E, G) is not in BCNF, find a BCNF decomposition of it. (d) Assume that (A, E, G) is decomposed into (A, G) and (E, G). Given the above functional dependencies, is this decomposition always lossless? If so, prove this. Otherwise, explain it using an example (i.e., a case of decomposing a table containg rows and columns into two tables). (e) Assume that R is decomposed into R1(A, B, C, D) and R2(A, E, G). Is this decomposition dependency preserving? Justify your answer.Consider the relation schema R(A, B, C, D, E, F) and the set S = {AB->C, BC->AD, D->E, C->B) of functional dependencies. Assuming that R is decomposed into the relation schemas R1(A,B), R2(B,C), R3(A,B,D,E) and R4(E,F). Use the Chase test to show if this is a lossless decomposition.
- Given: Relation R = {(a,a), (a.b), (b,a). (b,b). (c,c)} and Set S = {a, b, c} Complete Relation R to be Transitive with Set SSuppose that we have the following four tuples in a relation S with three attributes ABC: (1,2,3), (4,2,3), (5,3,3), (5,3,4). Which of the following functional (-->) and multivalued (-->>) dependencies can you infer does not hold over relation S? a) A-->Bb) A-->>Bc) BC-->Ad) BC-->>Ae) B->Cf) B-->>CA relation, R, on non-empty set A is defined as a subset of A × A: R ⊆ A × A For example, let A = {1, 2, 3} then Cartesian product A × A = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}. A binary relation R ⊆ A × A Briefly explain the meaning of the relation, providing an example (and diagram): (i)) Discuss the real-life applications of binary relations
- 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) keyCompute a canonical cover Fc of F = { A → E, BC → D, C → A, AB → D, D → G, BC → E, D → E, BC → A }, given a relation-schema R(A, B, C, D, E, G). If you can, please include the steps you use to reach with what Armstrong axioms you use to simplify. Please give proper explanation and typed answer only.Suppose R is a relation with attributes A1, A2, ..., An. As a function of n, tell how many superkeys R has, if The only keys are A1 and A2?
- Consider a relation with schema R(A,B,C,D,E,G) and functional dependencies (FDs) C→D; B→A; D→E; E→G ) What is the closure of {B,C}? Show steps of your solution. Find 2 nontrivial FDs that can be inferred from the given FDs set. Explain how you found them. Propose one key for a given schema and FDs. Explain how you found it.Consider a relation R(A,B,C,D,E) with functional dependencies AB→C, A→D, and D→E. (a) What are the keys for R?(b)What is the closure A, that is A+ ? What is the closure AB, that is (AC)+ ?. A relation R on a set S, defined as x R y if and only if y R x. This is an example of?a) reflexive relationb) symmetric relationc) transitive relationd) invalid relation