CMSC 203 - Discrete Structures - FINAL EXAMIMATION - Spring 2003
1. (a) Use the Laws of Logic to show: p ∨ (¬q → r) ≡ r ∨ (¬p → q)
(b) Find the negation of the following Universal Conditional:
For all integers, n, if n is odd, then n is not divisible by 2.
(c) Use the rules of inference to show the following is a valid argument:
p → s
r ∧ ¬s
(p ∨ t) → ¬r
∴ ¬t
2. (a) Find A × B for the sets A = {0, 00} and B = {1, 11, 111}
(b) Using the Properties of Sets, show that for any sets A and B, A ∪ (A ∩ B) = A
3. (a) Find the inverse of the function F = {(1,g), (2,d), (3,a), (4,b), (5,b)}.
(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(x+1) and G(x) = 3x − 5.
(d) Show that the function f: R → R given by f(x) = 3(x + 2) is One-To-One and Onto.
4. (a) Use the Euclidean Algorithm to find GCD(444,36).
(b) Find Big-O of the algorithm whose complexity is F(n) = [n7(log2 n) + n8 + 6n3](n5 + 3)
5. Prove 1 of the 2 Theorems below:
Theorem 1: For non-zero integers a, b, and c, if (a mod c) = (b mod c), then a ≡ b mod c.
Theorem 2: The square root of 2 is irrational.
6. (a) Find the next 3 terms of the sequence sn = (sn-1)(sn-2) when s0 = 1 and s1 = 4.
(b) Calculate the following series: (i) (ii)
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 < 4n for all n > 4.
8. (a) A restaurant serves 5 soups, 6 salads, 20 entrees, 10 desserts, and 15 beverages. How many din-
ners 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 8 have: (i) 3 ones? (ii) at least seven zeros?
(c) (i) How many ways can 6 suspects form a viewing line? (ii) If a certain pair must stand next to one
another?
(d) How many distinct piles of 100 coins (pennies, nickels, dimes, quarters, half-dollars, and dollars)
can I create from a vast quantity of coins, if I must have at least 5 of each in a given pile?
5i5i1–
–
i1=
100
∑5i
i0=
100
∑
Docsity.com
