Consider the relation courses (C, T,H, R,S, G), whose attributes may be thought informally as course, teacher, hour, room, student, and grade. Let the set of FD'S for courses be C\rightarrowT, HR\rightarrowC, HT\rightarrowR. HS\rightarrowR. CS\rightarrowG. 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 student. 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 a lossless-join, dependency-preserving decomposition of R into 3NF relations. Are any of the relations not in BCNF?