Consider the ER schema of relation F= (A, B, C, {D}, E, F). Knowing that there are the following (func- tional) dependencies: fd1: {A} → {C, E}, fd2: {A} → {D}, fd3: {A, B} → {F}. (a) ■Is F in 1NF? If "yes" why? If not modify F so as to put it in 1NF. Let us call F1 the result of this step (b) Is F1 in 2NF? If "yes" why? If not modify F1 so as to put it in 2NF. Let us call F2 the result of this step (c) Is F2 in 3NF? If "yes" why? If not modify F3 so as to put it in 3NF

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 the relation schema R (A, B, C, D, E, F, G, H) with the set of functional dependencies K = {A→B, B→DE, F→H, G→CE}. Determine all candidate keys of R in a systematic manner (do not use the Armstrong's Axioms) and explain how you determine them. Show each step.

Consider a relation schema R = {A, B, C, D, E} and its functional dependency set F = {A → BC,CD → E,B →D,E → A}. Let R be decomposed into R1 and R2, where R1 = (A,B,C) and R2 = (A,D,E). Is the decompositionlossless? Is the decomposition dependency preserving? (a) The decomposition is lossless and dependency preserving.(b) The decomposition is neither lossless nor dependency preserving.(c) The decomposition is not lossless but dependency preserving.(d) The decomposition is lossless but not dependency preserving.

Consider the relation R, which has attributes that hold schedules of courses and sections at a university; R = {{Course_no, Sec_no, Offering_dept, Credit_hours, Course_level, Instructor_ssn, Semester, Year, Days_hours, Room_no, No_of_student}. Suppose that the following functional dependencies hold on R: {Course_no} -> {Offering_dept, Credit_hours, Course_level}{Course_no, Sec_no, Semester, Year} -> {Days_hours, Room_no, No_of_students, Instructor_ssn} {Room_no, Days_hours, Semester, Year} -> {Instructor_ssn, Course_no, Sec_no}Answer the following: What set of attributes form the key of R? Normalize this relation to 3NF.

Computer Science Given the relation R (A, B, C, D, E, F, G) and the set of functional dependencies: F= {BCD → A, BC → E, A → F, F→G, C→D, A→G}, a) Decompose R into 3NF. Show the different steps. b) Is this decomposition also in BCNF? Why or why not?

We studied different properties of relations like reflexive, symmetric, antisymmetric, transitive. Please provide one example relation each for each property of the relation. Try to come up with original examples which can be seen practically. (Example, Reflexive relation R = {(a, b) | a and b share same birthday},  Symmetric relation R = {(a, b) | a and b are friends}, Antisymmetric relation R = {(a, b) | a is instructor of b}, Transitive relation R = {(a, c) | a is sibling of b, b is sibling of c, then a is sibling of c}

Consider a relation schema R = {M, N, P, Q, S} and its functional dependency set F = {N → Q,S → M,M →P,P → S}. Is MN a super key of R? Is MN a candidate key of R? (a) MN is not a super key for R; MN is not a candidate key for R.(b) MN is a super key for R; MN is not a candidate key for R.(c) MN is a super key for R; MN is a candidate key for R.(d) MN is not a super key for R; MN is a candidate key for R.

Let's suppose, a relation R=(A, B, C, D, E. H) with a set of functional dependencies F: (AE->BC, B ->AD, CD->E, E→CD, A->E) is given. Relation R (A, B, C, D, E, H) is decomposed into R1 (A, B, C, D) and R2 (A, B, E, H). Find out whether the decomposition is lossless or not, and whether it is dependency preserving or not.

1. Consider a relation schema R1(A, B, C, D) that satisfies the set of functional dependencies F1 ={A→B, B→C, C→A}.a) Calculate A + and (AC)+b) Can AB→ C? Justify your answer.c) Find all the candidate keys of R1 using F1 and show your steps to find them. 2. Consider a relation schema R2(A, B, C, D) that satisfies the set of functional dependencies F2 = {A→B, C→D, D→AC}.a) Is (AD) is candidate key of R2? Justify your answer.b) Is R2 in BCNF? Explain the reason to your answer.c) If R2 is not in BCNF, give a lossless-join BCNF decomposition of R2. Is this decomposition dependency preserving? 3. Consider a relation schema R3(A, B, C, D, E) that satisfies the set of functional dependencies F3 = {A → BC, AC → BD}.a) Calculate (AC)+ and (AE)+b) Is AB → C in F3 +? Justify your answer.c) Find the canonical cover Fc of F3.d) Is R3 in 3NF? Explain the reason to your answer. If R3 is not in 3NF, give a lossless-join, dependency preserving 3NF decomposition of R3.