CMSC 203 - Discrete Structures - FINAL EXAMIMATION - Spring 2005
1. (a) Use the Laws of Logic to show: q ∨ ¬ (r → p) ≡ (q ∨ r) ∧ (p → q)
(b) Find the negation of the following Universal Conditional:
All people who like Mathematics get good jobs.
(c) Use the rules of inference to show the following is a valid argument:
p → q r ∧ ¬q p r ∨ s ∴ s
2. (a) Find A × B for the sets A = {x, y} and B = {1, 2, 3}
(b) Using the Properties of Sets, show for any sets A, B and C, (A − B) − C = (A − C) − B.
3. (a) For F = {(b,g), (d,d), (g,a), (n,b), (r,b)}, what Domain and Image make F a function?
(b) Why or why not is the inverse in (a) a function?
(c) Find F(G(x)) for the following Real-valued functions: F(x) = 2(2x+1) and G(x) = 2x − 2.
4. (a) Use the Euclidean Algorithm to find GCD(154,84).
(b) Find Big-O of the algorithm whose complexity is F(n) = (3n5)(n6 + 5n4)(2n3 + 3n + 2).
5. Prove 1 of the 2 Theorems below:
Theorem 1: Show that the function f: R → R given by f(x) = π(x + e) is a bijection.
Theorem 2: The square root of 3 is irrational. (Assume the Lemma: If 3 divides INT2 then 3
divides INT)
6. (a) Find the next 3 terms of the sequence sn = 2sn-1 − 3sn-2 when s0 = 1 and s1 = 0.
(b) Find an expression for the series:
7. Prove 1 of the 2 Theorems that follow by Mathematical Induction.
Theorem 1: If a set has n elements, then it has 2n subsets.
Theorem 2: Is sn = sn-1 + sn-2 + sn-3 + sn-4 when s0 = s1 = s2 = s3 = 1 then sn is even, for all n > 4.
8. (a) A restaurant serves 4 soups, 6 salads, 8 entrees, 10 desserts, and 12 beverages. How many
dinners can they create if each dinner consists of a soup or salad, an entree, and a dessert or
beverage?
(b) How many binary strings of length 32 have no more than 3 zeros?
(c) How many ways can judges award 1st, 2nd, and 3rd Place prizes to 25 contestants?
(d) How many distinct piles of 200 coins (pennies, nickels, dimes, quarters, half-dollars, and dol-
lars) can I create from a vast quantity of coins, if I must have at least 10 of each in its pile?
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i
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