CMSC203 - Fall 2002 - Examination 1
1. Circle T for True or F for False as they apply to the following statements:
T F A conditional statement and its contrapositive are logically equivalent.
T F If a, b, and c are integers and a divides (b + c), then a divides b.
T F If F:X → Y and G:Y → X are functions, then F(G(y)) = G(F(x)).
T F Onto functions map bigger sets to smaller or equal-sized sets.
T F Disjoint sets have empty intersections.
T F The Integers and the Rationals have the same cardinality.
T F The Binary Search algorithm is of order O(2n).
T F If A and B are sets, then |A ∪ B| = |A| + |B| - |A ∩ B|.
T F If p and q are statements, then p → q ≡ ¬p ∨ q.
T F If P(x,y) is a predicate, then the negation of the quantified statement
For all x, there is a y such that P(x,y) is the quantified statement
For all y, there is an x such that P(x,y).
2. Find the truth table for the compound statement: p → ¬[q ∨ (r → ¬p)]
3. Find the related forms for the Universal Conditional Statement:
Every even integer greater than two is composite.
CONVERSE: ____ INVERSE: ____ CONTRAPOSITIVE: ____ NEGATION: ____.
4. Find set X so that the function, f:Z → X, given by f(n) = 2n is a bijection (one-to-one and onto).
5. (a) Find the first 4 terms of the sequence an = .
(b) For constant, c, and functions f and g, show that: .
6. (a) Rank from 1 (least complex) to 10 (most complex) the complexity of algorithms
with the following orders:
Order n2nlognn!2
n1nn
nlogn10nn10
Rank
(b) Find the Big-Oh of the algorithm with complexity: (n2 + 1)(n + n3) + (2n + 1)(n4).
7. (a) List the elements of a Bijective function from {1, 2, 3, 4, 5} to {2, -4, 8, -16, 32}
(b) Find the Inverse of the function you described in (a).
(c) Use Directed Graphs to verify that the composition of your function with its inverse is the
Identity function. Only test for the composition in one direction.
8. For the sets A = {1, 3, 4}, B = {2, 3, 4}, and C = {1, 2, 3}:
(a) verify that A ∩ (B − C) = B ∩ (A − C) (b) find A × B
9. Use the Laws of Logic to verify: (p → q) → ¬(p → ¬r) ≡ p ∧ (q → r).
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