CMSC 203 - Discrete Structures - FINAL EXAMIMATION - Part Two - Fall 2002
1. Construct the truth table for the compound proposition: [q ∨ (¬q → ¬p)] ↔ (p → ¬q)
2. Use the Laws of Logic to verify: p ∧ (p ∨ q) ≡ p.
3. What is the negation of the quantified statement: Every dog that chases parked cars has a flat nose.
4. Find A × B for the sets A = { ∅, {1} } and B = { {0}, ∅ }
5. Show that the function f: R → R given by f(x) = 9x + 7 is One-To-One and Onto.
6. Let the function f: R → R be f(x) = x + 3 and the function g: R → R be g(x) = x2 + 2x + 1. Find:
(a) f
(
g(x)) (b) g( f(x))
7. (a) Find the big-O estimate for the function F(n) = (n4log n + n6 + 6n)(n2 + 1).
(b) Find the Hamming Distance between 100110011001 and 101100111000.
(c) Using the Euclidean Algorithm, find gcd(1024,120).
8. If a, b, and c are integers with a = b + c, prove that gcd(a,b) = gcd(b,c).
(Hint: if gcd(a,b) ≤ gcd(b,c) and gcd(a,b) ≥ gcd(b,c), then gcd(a,b) = gcd(b,c)).
9. Use the following Theorem:
THEOREM: For all integers, n, and prime numbers, p, if p divides n2, then p divides n.
to prove by the Method of Contradiction that is irrational.
10. Let a be a Real Number not equal to 0 or 1. Use Math Induction to prove for all integers n > 0,
.
11. (a) How many ways can Art, Betty, Chad, Deb, Ed, Fran, Glen, Helen, Ivan, and Jane form a line if Ed can-
not be immediately between Betty and Ivan?
(b) The Mars Candy Company sells bags of M&M candies with 60 pieces candy colored from 8 different colors
in them. How many different bags can they produce if they want at least 1 of the first color, 2 of the second,
color, 3 of the third color, 4 of the fourth color, ..., and 8 of the eighth color?
3
ai
i0=
n
∑an1+ 1–
a1–
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