Exercise 3.5.1: For each of the relation schemas and sets of FD's of Exer- cise 3.3.1 i) Indicate all the 3NF violations. ii) Decompose the relations, as necessary, into collections of relations that are in 3NF Exercise 3.5.2: Consider the relation Courses(C,T, H, R, S,G), whose at- tributes may be thought of informally as course, teacher, hour, room, student, and grade. Let the set of FD's for Courses be C T, HR C, HT R, HS R, and CS G. Intuitively, the first says that a course has a unique teacher, and the second says that only one course can meet in a given room at a given hour. The third says that a teacher can be in only one room at a given hour, and the fourth says the same about students. The last says that students get only one grade in a course. a) What are all the keys for Courses? b) Verify that the given FD's are their own minimal basis c) Use the 3NF synthesis algorithm to find a lossless-join, dependency-pres- erving decomposition of R into 3NF relations. Are any of the relations not in BCNF? Exercise 3.5.3: Consider a relation Stocks(B, O,I,S, Q,D), whose attributes may be thought of informally as broker, office (of the broker), investor, stock, quantity (of the stock owned by the investor), and dividend (of the stock). Let the set of FD's for Stocks be S D, I+ B, IS Q, and B O. Repeat Exercise 3.5.2 for the relation Stocks. Exercise 3.5.4: Verify, using the chase, that the decomposition of Exam- ple 3.27 has a lossless join. !! Exercise 3.5.5: Suppose we modified Algorithm 3.20 (BNCF decomposition) so that instead of decomposing a relation R whenever R was not in BCNF, we only decomposed R if it was not in 3NF. Provide a counterexample to show that this modified algorithm would not necessarily produce a 3NF decomposition with dependency preservation. 3.6 Multivalued Dependencies A "multivalued dependency" is an assertion that two attributes or sets of at- tributes are independent of one another. This condition is, as we shall see, a generalization of the notion of a functional dependency, in the sense that