Exam 2 - Discrete Structures - Spring 2008
1. Circle T if the corresponding statement is True or F if it is False.
TFThe sequence {1, โ1, 1, โ1, 1, โ1 ...} is an example of an Alternating sequence.
TFThe Weak and Strong Principles of Mathematical Induction are logically equivalent.
TFIn general, iterative algorithms use memory more efficiently than their equivalent
recursive representations.
TF1 + 10 + 100 + 1000 + ... + 10n = 10(n + 1) โ 1.
TF2 + 4 + 6 + 8 + 10 + ... + 2,000 = 1,001,000.
TFAlgorithms whose order is O(n2) are less efficient than those of order O(2n).
TFIf the set A contains the element x, then half the subsets of A contain x.
TFLCM( p,q)GCD( p,q) = pq.
2. Let {an} and {bn} be the sequences defined, for n > 0, by:
an = (n โ 1), and bn = (n + 1). Find c1, c2, c3, and c4 when cn = (an)(bn).
3. Write out the Euclidean Algorithm and trace its steps to calculate GCD(21,4).
4. (a) Give a Recursive Definition for the odd integers.
(b) Find the Big-Oh of the algorithm with complexity: (n2 + 1)(4n3 + 5) + [2n4 + n(log n)](3n2).
5. (a) Given a = 22335577119131117131915231729193121. and b = 21325478111613321764191282325629512311024,
GCD(a,b) = ___________________ LCM(a,b) = __________________________.
(b) List out the search intervals in applying the Binary Search algorithm to find 7 in the list:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(Hint: Include the mid/test point with the lower interval.)
6. Prove one of the two Theorems below using Mathematical Induction.
Theorem 1: For all integers n > 0, .
Theorem 2: If a0 = 0, a1 = 10, and a2 = 20, then an = anโ1 + anโ2 + anโ3 is divisible by 10,
for all n > 3.
7. Prove one of the two Theorems below:
Theorem 1: If a, b, p, and r are Integers with a = bq + r, then GCD(a,b) = GCD(b,r).
Theorem 2: The product of odd integers is odd.
8. Prove one of the two Theorems below by Contradiction or Contraposition.
Theorem 1: is irrational.
Theorem 2: The Prime numbers form an infinite set.
3i
i0=
n
โ3n1+ 1โ
2
---------------------=
2
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