Show that it is possible to ensure that a dependency-preserving decomposition into 3NF is a lossless decomposition by guaranteeing that at least one schema contains a candidate key for the schema being decomposed. (Hint: Show that the join of all the projections onto the schemas of the decomposition cannot have more tuples than the original relation.)

Consider the relation R = {A, B, C, D, E, F, G, H, I,J} and the set of functional dependencies F = { {A, B} → {C}, {A} → {D, E}, {B} → {F}, {F} →{G, H}, {D} → {I, J} }. (b) Whether (A,B,F} is a super key? Whether {A,B,F} is a candidate key? Why? (c) Whether {A,D} is a super key? Whether {A,D} is a candidate key? Why?

Suppose you are given a database relation R with single-valued atomic attributes (A, B, C, D, E) and a set of functional dependencies F = {AB-C, AC→B, AE, B→D, BC→AE} (a) List all candidate keys for R. (6 points) (b) Name the strongest normal form that holds on R and explain. (4 points) (c) Decompose it into a collection of BCNF relations if it is not in BCNF. Show all steps. (6 points) For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). Open Sans,ari... 10pt x² X2 می P> † {;} Ка ? لاک (+ BIUS Paragraph == = 由用 EXE + Œ P - ||| ABC A П "" Ω Ix 用 : 0 WORDS POWERED BY TINY

The division operator of relational algebra, “÷”, is defined as follows. Let r(R)and s(S) be relations, and let S ⊆ R; that is, every attribute of schema S isalso in schema R. Given a tuple t, let t[S] denote the projection of tuple t onthe attributes in S. Then r ÷ s is a relation on schema R − S (that is, on theschema containing all attributes of schema R that are not in schema S). A tuplet is in r ÷ s if and only if both of two conditions hold:• t is in ΠR−S(r)• For every tuple ts in s, there is a tuple tr in r satisfying both of the following:a. tr[S] = ts[S]b. tr[R − S] = tGiven the above definition:a. Write a relational algebra expression using the division operator to findthe IDs of all students who have taken all Comp. Sci. courses. (Hint:project takes to just ID and course id, and generate the set of all Comp.Sci. course ids using a select expression, before doing the division.)b. Show how to write the above query in relational algebra, without usingdivision. (By doing so, you…