SIf a decomposition of a relation R into R¡ and R2 is lossless, then which of the followings are true? |(a) R can be reproduced by natural join of R1 and R2 (i.e, by R¡ ¤R2).. |(b) All the functional dependencies can be checked without computing Rị ¤ R2. (c) R1NR2 forms a super key for either R1 or R2. |(d) All of them.

For each of the following pairs of expressions, give instances of relations thatshow the expressions are not equivalent. ΠA(r − s) and ΠA(r) − ΠA(s).

Solve the following:

Solve the following:

Give all the keys of R if the functional dependencies on R in each of the following cases

2.Consider relation R=(A,B,C,D) with the following FDs: B → C, C > D, and D A a. List all candidate keys of R. b. Is R in 3NF? BCNF? Justify your answer

QUESTION 6 Consider the following functional dependencies for a relation R(A, B, C, D) A -> (B, C, D), (A, D) -> (B, C) (C, D) -> (A, B) Identify the candidate key(s). O {C, D} O {A}, {C, D}, {A, D} O {A}, {C,D} O A

21. of 40 A decomposition is in 3NF if every decomposed relation is in 3NF. A decomposition is in BCNF if every decomposed relation is in BCNF. Let R(A,B,C,D,E,F,G) be a relation with the functional dependencies: C→ E, EG→ D, DG-→A, and A→ BF. If we decompose R into relations R1(C, D, G), R2(A, D, G), R3(A, B, F), and R4(C, E), which of the following statements is TRUE? Select one: O The decomposition is in BCNF but Not in 3NF. The decomposition is in 3NF and in BCNF. The decomposition is in 3NF but Not in BCNF. O The decomposition is Not in 3NF and Not in BCNF.

MC 2  Write a Python function that determines whether a given relation R on S = {1, 2, 3, 4} is transitive but not reflexive. Then apply this functions to all relations R on S to count how many of them are transitive but not reflexive. Show your program and the number of transitive but not reflexive relations on S it found. .   THE COUNT IS IMPORTANT

Implement in Python / Java Algorithm: Testing for lossless (nonadditive) join property. Input: A universal relation R, a decomposition D = { R1, R2, R3, ….. Rm } of R, and a set F of functional dependencies. 1. Create an initial matrix S with one row i for each relation in Ri in D, and one column j for each attribute Aj in R. 2. Set S(i, j) := bij for all matrix entries. (* each bij is a distinct symbol associated with indices (i, j) * ) {for each column j representing attribute Aj                                  {if (relation Ri includes attribute Aj ) then set S(I, j):=aj;};};             (* each aj is a distinct symbol associated with index (j) *)   3. For each row i representing relation schema Ri {for each functional dependency X  → Y in F            {for all rows in S which have the same symbols in the columns               corresponding to attributes in X                  {make the symbols in each column that correspond to                                 an attribute in Y be…