CMSC 203 - Discrete Structures - Fall 2002 - Examination 3
1. Show the sequence {an} defined by an = 10n − 7 satisfies the recurrence relation: an − 2an−1 + an−2 = 0.
2. Find the General Solution to the following recurrence relations:
(a) an = 7an−1 + 18an−2(b) an = 18an−1 − 81an−2
3. Find the Particular Solutions to the following General Solutions:
(a) an = A(5)n + B(−2)n, when a0 = 4 and a1 = 13 (b) an = (A + Bn)(−15)n, when a0 = −2 and a1 = 15
4. (a) Validate that R = {(a,b) | a,b are Integers and (a + b) is even} is an Equivalence Relation
(b) For this relation, find [1].
5. For the collection of records using the keys (Name, Gender, Age, Residence, Citizenship):
{(Mary, Female, 32, New York, USA), (Louise, Female25, Manchester, UK),
(Timothy, Male, 42, Dallas, USA), (Albert, Male, 32, Paris, French),
(Elsa, Female, 27, Cologne, German), (Victor, Male, 42, Toronto, Canada)}
(a) which keys are Primary Keys? (b) what new records constitute P2,3?
6. Find MR o MR for the relation, R, whose matrix representation is given by:
MR =
7. List the elements of the Equivalence Relation, R, that induces the partition
{ {1}, {2,3}, {4,5,6} } on the set {1, 2, 3, 4, 5, 6}.
8. Find the Disjunctive Normal Form (Sum of Products) for the Boolean Polynomial F(w, x, y, z) = wx(y’ + z)
9. Find the Truth Table of the Boolean Polynomial whose Disjunctive Normal Form is
F(x, y, z) = xy’z + xyz’ + x’yz + x’y’z + x’yz’.
10. For the Boolean Polynomial, F(x,y) = xy’ + x’y, show that CNF(F) = [DNF(F’)]’.
1100
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