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CS 4320/5320 Homework 3Fall 2016Due Sun Oct 30, 2016, 11:59pmThis assignment is due 30 October 2016. It is out of 100 points and counts for 10% of your overall grade.1Functional Dependencies (20 Points)Prove each of the statements below. Remember that if your statement is an “if and only if” you need toprove both directions.(a) (6 Points) LetX,Ybe sets of attributes of a relation. Prove (X)+= (Y)+if and only ifX→YandY→X.(b) (6 Points) Prove that relationRsatisfiesX→Yif and only ifXis a superkey ofπXY(R).(c) (8 Points) LetXandYbe sets of attributes withX∩Y=Zand assumeXis nonempty. Prove thatfor any relationRwhose attribute set is exactlyX∪Y, ifZ→Yholds onR, thenR=πX(R)./ πY(R).2Normal Forms (30 Points)Consider a relation R with attributesABCDEFGHand the following functional dependencies:•B→CDF•A→EH•CDE→BGF(a) (8 points) Find all keys of Rwithoutiterating over all possible subsets of attributes. Explain yourprocess.(b) (8 points) R is neither in BCNF nor in 3NF. For each of these normal forms, explain why R does notsatisfy it.(c) (14 points) Find a decomposition of R into 3NF that is both lossless-join and dependency-preserving.Write down your reasoning. (See Section 19.6.2 in your textbook for details of the algorithm(s) to use.)1

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