Let R be an equivalence relation on A = {0, 5, 7,9} and R = {(0, 9), (0, 5), (7, 7), (9,5), (5, 9), (9, 9), (5,0), (0, 0), (9, 0), (5,5)}. Answer each of the following questions. [5] = 0 [9] = [0] [7]

give a recurrence relation that describes the sequence, answer both boxes.

4. Draw the Hasse Diagram for the following partial orderings.(a) S = {a, b, c} and ρ = {(a, a),(b, b),(c, c),(a, b),(b, c),(a, c)}(b) S = {a, b, c, d} and ρ = {(a, a),(b, b),(c, c),(d, d),(a, b),(a, c)}(c) S = {∅, {a}, {a, b}, {c}, {a, c}, {b}} and AρB ↔ A ⊆ B.

Consider the relation R on Z? given by: R = {((a, b), (c, d)) : a +d = c+b} Prove that R is an equivalence relation, and determine 3 elements of [(1,3)], the equivalence class of (1, 3).

Let R be a relation on A = {0, 1, 6, 8} and R = {(0,0), (8, 6), (6, 6), (8, 8), (1, 1), (1,0), (0, 1)}. Are the following statements true or false? 1. R is symmetric ? ? ? ? ? ? V V 2. R is transitive 3. R is reflexive 4. R is an equivalence relation 5. R is anti-symmetric 6. R is a partial order

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.

Identify each relation on N as one-to-one, one-to-many, many-to-one, or many-to-many. (a) R = {(1, 6) , (1, 4) , (1, 6) , (3, 2) , (3, 4)}(b) R = {(12, 5) , (8, 4) , (6, 3) , (7, 12)}(c) R = {(2, 7) , (8, 4) , (2, 5) , (7, 6) , (10, 1)}(d) R = {(9, 7) , (3, 4) , (3, 6) , (2, 4)}

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?

answer the following boxes

1) Consider the following relation instance of R1(A, B, C) contains the tuples {(a1,b1.c1). (a2,b1,c1), (a3,b3,c3), (a4,b3,c3), (a5,b5.c5). (a6,b6,c6)} and S1(B, C, D, E) contains the tuples ((b1.c1,d1,e1), (b1.c1.d2.e2),.(b1.c1,d3,e3), (b2.c2,d2.e2), (b3,c3,d4,e4).(b6.c7.d8,e9)) The following SQL query will result in how many tuples? Select From R1 r LEFT OUTER JOIN S1 s (ON r.B= s.B AND r.C=s.C): (The answer will simply be an integer, like 25). Answer: 2) Consider a relation schema R = (A, B, C, D, E, F, G} with the following set of functional dependencies: F = (BG A, CGA, A BC, BC A, BD E, EF, CF B) The decomposition of R into R1 = (A, B, C, G), R2 = {B, D, E), R3 = (B, C, F) and R4 = (E, F) is O In BCNF but Not in 3NF. In 3NF but Not in BCNF. In 3NF and in BCNF. Not in 3NF and Not in BCNF 3)