Consider a relation R(A, B, C, D, E, F), with FDs AB → C , BC → D, D → E, BE → F and DF → C. 1. Is R in BCNF? Explain why or why not. 2. Find the closure of the following: 1. (A, B) 2. (B, D) 3. (D, F) 3. List one superkey of R that is not a (minimal) key

5 of 40 A decomposition is in 3NF if every decomposed relation is in 3NF. A decomposition is in BCNF if every decomposed relation is in BCNF. Let R(A,B,C,D,E,F,G) be a relation with the FDs: F = {CD → G, DG → C, AD→ G, G –→A, AB → F}. Consider the decomposition of R into R1(C, D, G), R2(A, D, G), R3(A, B, F) and R4(C, E). Which of the following statements is TRUE about this decomposition? Select one: O Ris in BCNF but Not in 3NF. Ris in 3NF but Not in BCNF. R is Not in 3NF and Not in BCNF. O R is in 3NF and in BCNF.

List all 3-tuples in the relation {(a,b,c)| abc = 6} where a, b and c are all positive integers.

a) Define a relation R on R (the set of all real numbers) as follows: For all x, y E R, xRy → x < y. Is R reflexive, symmetric and transitive?

Consider an equivalence relation R on X={a,b,c}. R= {(a,a), (a,c), (b,b), (c,a), (c,c) } [a]=

Consider F = {BC -> D, B -> E, CE -> D, E -> CA, BF -> G} and R(A,B,C,D,E,F,G) 1.Find a minimal cover for F. 2.Find all many keys for R.

1. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is, for all (x, y) ≤ A × B, xSy ↔ x|y. State explicitly whic ordered pairs are in S and S-1

5. Let S = {a,b,c,d}, T = {a, c, e, f}, R = {c, f}. Find the following sets.... (a) SUT (b) SnT (c) SU (TNR) (d) T-R (e) Sketch a Venn diagram showing how S, T, and R relate to each other. (f) Let f: R→ R be given by f (x) = 3x - 2. Prove that f (x) is one-to-one and onto.

The domain for the relation R is the set of all integers. For any two integers, x and y, xRy if x evenly divides y. An integer x evenly divides y if there is another integer n such that y = xn. (Note that the domain is the set of all integers, not just positive integers.) Symmetric Transitive Anti-symmetric Reflexive

FInd the relation R that induces the following partition of set A: {{1,5},{2,3,6}, {4}}.