Download CSE 232A Midterm Winter 2007

CSE 232A Midterm Winter 2007
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You have 80 minutes for this open-book and open-slides exam. Good luck!
Problem 1 (20 points) True or false (no justification required):
1. If R and S are over attributes AB,
πA (R − S) = πA (R) − πA (S)
-Answer:
2. If R and S are over attributes AB,
σA=0 (R − S) = σA=0 (R) − σA=0 (S)
-Answer:
1
3. If R and S are over attributes AB,
πA (R ∪ S) = πA (R) ∪ πA (S)
-Answer:
4. If R and S are over attributes AB,
πA (R ∩ S) = πA (R) ∩ πA (S)
-Answer:
5. If R has attributes AB and S1 , S2 have attribute B,
R ÷ (S1 ∩ S2 ) = (R ÷ S1 ) ∩ (R ÷ S2 )
-Answer:
6. If R has attributes AB and S1 , S2 have attribute B,
R ÷ (S1 ∪ S2 ) = (R ÷ S1 ) ∪ (R ÷ S2 )
-Answer:
Given a unary operator o taking a set as argument and returning a set,
we say that o is monotonic if for every pair of sets S1 , S2 , whenever S1 ⊆ S2 ,
we have o(S1 ) ⊆ o(S2 ). If o is a binary operator from pairs of sets to sets,
we call o monotonic in its first argument if for every sets S1 , S2 , S, whenever
S1 ⊆ S2 , we have o(S1 , S) ⊆ o(S2 , S). o is monotonic in its second argument
if for every sets S1 , S2 , S, whenever S1 ⊆ S2 , we have o(S, S1 ) ⊆ o(S, S2 ).
7. The RA selection operator σ is monotonic.
-Answer:
2
8. The RA projection operator Π is monotonic.
-Answer:
9. The RA Cartesian product operator is monotonic in both arguments.
-Answer:
10. The RA set difference operator − is monotonic in both arguments.
-Answer:
3
Problem 2 (10 points)
Given relation R(A, B, C), express the fact that A, B is a key on R using
an algebraic dependency of the kind
E1 ⊆ E 2
where E1 , E2 are valid RA expressions. That is, denoting with E(D) the
result of applying query E on database D, find E1 and E2 such that for
every database D (which contains a relation R), the above key constraint
holds if and only if E1 (D) ⊆ E2 (D).
You are not allowed to use ∅ as one of the RA expressions. Moreover, E1 , E2 cannot be unsatisfiable queries. That is, there must exist some
databases D1 , D2 such that the result of E1 on D1 is non-empty, and so is
that of E2 on D2 .
4
Problem 3 (40 points) Consider the following schema of a boat rental
agency:
sailor(sname:string, rating: integer)
boat(bname:string, color: string, rating: integer)
reservation(sname:string, bname: string, day: date)
The rating attribute for boats indicates the minimum rating required of a
sailor to qualify for sailing (and therefore reserving) that boat.
Find the names of the highest-rated boats reserved by Popeye.
Express this query in (i) tuple calculus, (ii) SQL, (iii) domain calculus, (iv)
QBE, (v) relational algebra.
–Answer:
(i)
(ii)
(iii)
5
(iv)
(v)
6