Information Systems for Engineers. Exercise 5. ETH Zurich, Fall Semester Hand-out Due

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1 Information Systems for Engineers Exercise 5 ETH Zurich, Fall Semester 2017 Hand-out Due Reading material: Chapter 2.4 in [1]. Lecture slides Given the two tables below, write down the resulting table for each of the following operations. gini gini Panama 32.7 Switzerland 29.5 A = B = Georgia 40.9 Costa Rica 40.7 Costa Rica 40.7 Iran 37.4 (a) A B gini Panama 32.7 Georgia 40.9 Costa Rica 40.7 Switzerland 29.5 Iran 37.4 (b) A B gini Costa Rica 40.7 (c) A B gini Panama 32.7 Georgia 40.9 (d) B A gini Switzerland 29.5 Iran

2 (e) π gini (A) gini (f) σ gini 30 (B) gini Costa Rica 40.7 Iran 37.4 (g) π (σ gini<40 (A B)) Panama Switzerland Iran (h) B σ gini<40 (A) gini (i) σ gini>40 (A) B gini Georgia Given the two tables below, write down the resulting table for each of the following operations. id time in 1 Wang 10:03:33 true C = D = 2 Li 10:36:01 false 3 Zhang 14:02:57 true (a) C D Page 2

3 id time in 1 Wang 10:03:33 true 1 Wang 10:36:01 false 1 Wang 14:02:57 true 2 Li 10:03:33 true 2 Li 10:36:01 false 2 Li 14:02:57 true 3 Zhang 10:03:33 true 3 Zhang 10:36:01 false 3 Zhang 14:02:57 true (b) π last (ρ last (C)) last Wang Li Zhang (c) π enter (σ in=true (ρ time enter (D))) π (C) enter 10:03:33 Wang 14:02:57 Wang 10:03:33 Li 14:02:57 Li 10:03:33 Zhang 14:02:57 Zhang (d) γ id,,count(time) count (C π time (D)) id count 1 Wang 3 2 Li 3 3 Zhang 3 3. Given the two tables below, write down the resulting table for each of the following operations. birthdate Ryan Gosling E = Harrison Ford Sean Young Page 3

4 F = (a) E F movie title movie year star Arrival 2016 Amy Adams La La Land 2016 Ryan Gosling La La Land 2016 Emma Stone Bladerunner 1982 Rutger Hauer Bladerunner 1982 Harrison Ford Bladerunner 1982 Sean Young Bladerunner Ryan Gosling Bladerunner Harrison Ford Alien 1979 Sigourney Weaver No matching columns: cannot join tables. (b) E ρ star (F ) birthdate movie title movie year Ryan Gosling La La Land 2016 Harrison Ford Bladerunner 1982 Sean Young Bladerunner 1982 Ryan Gosling Bladerunner Harrison Ford Bladerunner (c) γ birthdate,count(movie title) movie count (ρ star (F ) E) birthdate movie count Consider the two tables below. For each of the following statements, write a corresponding relational algebra expression: G = hdi gini Costa Rica USA Switzerland Georgia Panama Indonesia H = population area Panama Georgia Costa Rica Indonesia Switzerland USA (a) The and Gini-index of countries with a population less than five million. π,gini (G σ population< (H)) (b) The and population of countries with a Gini-index of at most 40. π,population (σ gini 40 (G) H) Page 4

5 (c) The total number of people living in countries with an HDI of less than 0.8. γ SUM(population) total population (π population (σ hdi<0.8 (G) H)) 5. Given the two tables below, write down the resulting table for each of the following operations assuming bag semantics. I = (a) I J Frost Lich Jaina Baron Geddon Playable hero J = Open The Waygate Arcane Giant Molten Reflection Quest Frost Lich Jaina Baron Geddon Open The Waygate Arcane Giant Molten Reflection Playable hero Quest (b) I J (c) I J Frost Lich Jaina Baron Geddon Playable hero (d) J I Open The Waygate Arcane Giant Molten Reflection Quest (e) σ = (I J) Page 5

6 Molten Reflection (f) δ(σ = (I J)) Molten Reflection (g) π (σ count=b (a)), where a = γ,count() count (I J) b = γ MAX(count) max (π count (a)) You can assume that single-row, single-column tables are automatically unboxed to a scalar value. 6. The set of tables T can be defined as: T = {S P(S V) t, u S, support(t) = support(u)} Note that P(R) denotes the powerset of R. The notation (R) can be used interchangeably. The selection operator may be defined as: σ φ : T T σ φ (R) = {t (t R) φ(t)} = {t (t R) (t φ 1 (true))} = R φ 1 (true) (1) The selection operator is commutative: σ ψ (σ φ (R)) = (R φ 1 (true)) ψ 1 (true) = R (φ 1 (true) ψ 1 (true)) = R (ψ 1 (true) φ 1 (true)) = σ φ (σ ψ (R)) (2) Prove the following: (a) Projections commute with one another. Page 6

7 π A (π B (R)) = π A ( t R{t B } ) = {t B A } = B A } t R t R{t = {t A B } = A B } t R t R{t ( ) = π B A } = π B (π A (R)) t R{t (3) (b) Formally, a selection is defined only when the attributes used in its predicate φ can be found in the relation/table it is applied to. Find a condition that must hold so that both σ φ (π A (R)) and π A (σ φ (R)) are defined. Assuming that this condition holds, prove that these two expressions are equal, i.e. selections commute with projections. Selections commute with projections when the attributes used in the selection predicate are a subset of the attributes used in the projection. σ φ (π A (R)) = σ φ ({t A t R}) = {t A t R φ(t)} = {t A t R φ(t A )} π A (σ φ (R) = π A ({t t R φ(t)}) = {t A t R φ(t)} = {t A t R φ(t A )} (4) (5) (c) Theta joins commute with one another. R θ S = σ θ (R S) = σ θ (S R) = S θ R (6) The cartesian product of two sets is generally not commutative. However, in relational algebra the cartesian product of two tables is, because we do not care about the order of the attributes! Note: You may assume that union and intersection are commutative and associative. 7. Selection is distributive over union: σ φ (R S) = (R S) φ 1 (true) = (R φ 1 (true)) (S φ 1 (true)) = σ φ (R) σ φ (S) (7) (a) Prove that selection is distributive over intersection. Page 7

8 σ φ (R S) = (R S) φ 1 (true) = R S φ 1 (true) φ 1 (true) = (R φ 1 (true)) (S φ 1 (true)) = σ φ (R) σ φ (S) (8) (b) Prove that selection is distributive over set difference (subtraction). Hint 1: A B = A \ B C Hint 2: It might be easier to prove that σ φ (R \ S) = A and that σ φ (R) \ σ φ (S) = A, where A is an intermediate expression. σ φ (R \ S) = {t (t R \ S) φ(t)} = {t (t R) (t / S) (t φ 1 (true))} = {t (t R) (t S C ) (t φ 1 (true))} = R S C φ 1 (true) = R φ 1 (true) S C = σ φ (R) S C = σ φ (R) \ S σ φ (R) \ σ φ (S) = σ φ (R) (σ φ (S)) C = σ φ (R) (S φ 1 (true)) C = σ φ (R) (S C φ 1 (false)) = (σ φ (R) S C ) (σ φ (R) φ 1 (false)) = (σ φ (R) \ S) (R φ 1 (true) φ 1 (false)) = (σ φ (R) \ S) (R Ø) = (σ φ (R) \ S) Ø = σ φ (R) \ S (9) (10) 8. The projection operator may be defined as: π A : T T π A (R) = {t A t R} = t R{t A } (11) Note that, if t is a tuple, then: t A=α1,α 2,...,α n = (α 1 t.α 1, α 2 t.α 2,..., α n t.α n ) The projection operator is distributive over union: Page 8

9 π A (R S) = {t A } = t R S {t A } t {r 1,...,r m,s 1,...,s n} = {r 1 A }... {r m A } {s 1 A }... {s n A } = {t A } A } t R t S{t (12) = π A (R) π A (S) (a) Is the projection operator distributive over intersection? If yes, prove it, otherwise provide a counterexample. π A ({(A α, B β)} {(A α, B β )}) =π A (Ø) =Ø π A ({(A α, B β)}) π A ({(A α, B β )}) =π A ({(A α)}) π A ({(A α)}) =π A ({(A α)}) ={(A α)} (13) (b) Is the projection operator distributive over set difference (commonly denoted as or \ )? If yes, prove it, otherwise provide a counterexample. π A ({(A α, B β)} \ {(A α, B β )}) =π A ({(A α, B β)}) ={(A α)} π A ({(A α, B β)}) \ π A ({(A α, B β )}) =π A ({(A α)}) \ π A ({(A α)}) ={(A α)} \ {(A α)} =Ø (14) 9. Prove that a projection is absorbed by a projection on a smaller subset. π A (π A,B (R)) = π A ( t R{t A,B } ) = t R{t A,B A } = t R{t A } (15) = π A (R) Page 9

10 References [1] Garcia-Molina, Hector and Ullman, Jeffrey D. and Widom, Jennifer, Database Systems: The Complete Book, Second Edition. Page 10