Download Discrete Mathematics Practice Questions and more Exams Discrete Mathematics in PDF only on Docsity!
Discrete Mathematics FAMAT State Convention 2004
For all questions, E. NOTA means none of the above answers is correct.
- Paul would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Noah is not the highest paid of the three, then Graham is. Second, he knows that if Graham is not the lowest paid, then John is the most. What is the order of decreasing relative salary?
A. Noah, Graham, John D. Cannot be determined B. John, Noah, Graham E. NOTA C. Noah, John, Graham
- The logic function, F , is a function of three inputs, A , B , and C. F takes on the values shown in the
table for the given values of A , B , and C. (Note: A denotes the complement of A, A+B denotes A OR B, A ⋅ B denotes A AND B.)
A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
Which of the following is an expression for F?
A. F =(^ A ⋅ B ⋅ C )^ +(^ A ⋅ B ⋅ C )
B. F =( A ⋅ B ⋅ C ) +( A ⋅ B ⋅ C )
C. F =(^ A ⋅ B ⋅ C )^ +(^ A ⋅ B ⋅ C )
D. F =( A ⋅ B ⋅ C )
E. NOTA
- On a sunny Florida afternoon, Alexander proclaims, “I am the best rapper in Gainesville and I also love to eat chicken wings.” What is the negation of his statement?
A. “I am not the best rapper in Gainesville and I do not love eating chicken wings.” B. “I am not the best rapper in Gainesville or I do not love eating chicken wings.” C. “If I am the best rapper in Gainesville, then I do not love eating chicken wings.” D. The statement is not a proposition, so it cannot be negated E. NOTA
- During a discussion with her parents, Robin declares, “If I get accepted to Georgia Tech, then I will study Political Science.” What is the converse of her statement?
A. “If I don’t choose to study Political Science, I will not get accepted to Georgia Tech.” B. “If I choose to study Political Science, I will get accepted to Georgia Tech.” C. “I will not get accepted to Georgia Tech and I will not study Political Science.” D. “I will get accepted to Georgia Tech or I will study Political Science.” E. NOTA
- Which of the following is logically equivalent to P + ( P ⋅ Q )? (Note: A denotes the complement of
A, A+B denotes A OR B, A ⋅ B denotes A AND B.)
A. P ⋅ Q D. P ⋅ Q
B. ( P ⋅ Q ) +( P ⋅ Q ) E. NOTA
C. T
- The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. Which of the following functions map N (the set of positive integers) to the set of odd positive integers, thus showing equal cardinality?
A.
n f n = D. f ( n ) = 2 n − 1
B. f ( n )= 2 n + 1 E. NOTA C. f ( n )= n^2 − 1
- A set that is either finite or has the same cardinality as the set of natural numbers is called countable. A set that is not countable is called uncountable. How many of the following sets are, in fact, countable?
i. All possible outcomes of the Georgia State lottery ii. Real numbers between -1 and 1 (not inclusive) iii. Integers greater than 5 and divisible by 2 iv. Rational numbers between 100 and 100. v. Complex numbers, z , that satisfy z ≤ 1
A. 1 B. 2 C. 3 D. 4 E. NOTA
- An algorithm for finding the maximum element in a finite sequence of integers is given below. What is the time complexity of this algorithm?
A. O(1) B. O(n) C. O(log n) D. O(n^2 ) E. NOTA
procedure maximum(a 1 ,a 2 ,…,an: integers) max := a 1 for i := 2 to n if max < ai then max := ai
end with max as largest value
- The prime factorization of 7007 can be written as with A < B < C. What is the value of ?
A^2 ⋅ B ⋅ C
2 A + B − C
A. 11 B. 17 C. -3 D. 13 E. NOTA
- A function Φ ( n )is defined recursively for nonnegative integers n as follows:
n (n-1)
( n ) whenn 0
whenn 0
What is the value of Φ ( 7 )?
A. 720 B. 120 C. 49 D. 64 E. NOTA
- Among 100 people, at least how many people were born in the same month?
A. 8 B. 9 C. 10 D. 11 E. NOTA
- How many ways are there to select a committee consisting of 3 physics professors and 4 math professors if there are 9 total physics professors and 11 total math professors?
A. 414 B. 27,720 C. 1140 D. 330 E. NOTA
- Simplify the following expression: (^) with n and
− k
n k
n 1
≥ k n , k ∈ Z +
A. (^) ^ B. (^) C. n D. (^) E. NOTA
k − k
n 2
2
k
n 1 ! (^)
k + 1
n
- What is the probability that a bit string of length four, generated at random, contains at least two consecutive zeros given that the first bit is a zero?
A.
B.
C.
D.
E. NOTA
- Richard invented a gambling game in an attempt to win money off of his younger cousins. He will roll a single die one time and take note of the result. If the die shows an odd number, Richard must pay $1, if the die shows a 2, nothing happens, and if the die shows anything else, Richard wins $2. What is the average amount of winnings for Richard on each roll? (Approximate to the nearest cent)
A. $0.50 B. $1.00 C. $0.17 D. $0.25 E. NOTA
- A simple graph is shown to the right. What is the adjacency matrix for this graph with respect to the ordering of vertices A,B,C,D?
A. B.
C.
D. E. NOTA
- A directed graph is shown to the right. What is the adjacency matrix for this graph with respect to the ordering of vertices A,B,C,D?
A. D.
B. E. NOTA
C.
- A simple graph is shown to the right. Does this graph have a Euler circuit?
A. Yes B. No C. Euler circuits do not apply to simple graphs D. Cannot be determined E. NOTA
- How many of the following statements are true?
i. if n is an odd integer, then n^2 is an odd integer ii. If 3n+2 is an odd integer, then n is an odd integer iii. 2 is irrational iv. The integer n is odd if and only if n^2 is odd
A. 0 B. 1 C. 4 D. 3 E. NOTA
- What is the least common multiple of 2 3 35 72 and 2 4 33?
A. 2^3 B. 2^43572 C. 2^333 D. 7^2 E. NOTA
