discrete mathematics

— – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

PART A

Each question is worth 2 points.

1. m mod n will have values ranging from 0 to n.

T or F

2. -39 MOD 7 and -37 MOD 5 are congruent.

T or F

3. The base system of the value 275 must be either decimal or

octal.

T or F

4. The value 967 is a prime number.

T or F

5. The value 659 has a maximum of 25 possible prime factors

Because the square root of 659 is 25+.

T or F

6. A Permutation of the elements of a set is an ordered

arrangement of the elements of the set.

T or F

7. P(6,4) = 360

T or F

8. C(7,5) = 21

T or F

9. Consider the following relations on {1, 2, 3 } :

R1 = { (1,1), (2,2), (3,3) }, and

R2 = { (1,2), (1,3), (3,2) }.

R1 is symmetrical and R2 is transitive

T or F

10. Using members of the set {1, 3, 4, 5, 7, 8}, the next

larger P(6,3) permutation after 431 is 454.

T or F

11. The Sum Rule is applied when the tasks to be performed

are disjoint.

T or F

12. According to the Pigeonhole principle, when (m+12) items are

to be placed in (m+7) boxes, there will be more than one item

in at least one box.

T or F

13. Pascal’s Triangle yields the value of the coefficients of an

algebraic expansion.

T or F

14. The probability of picking a “face” card (Jack, Queen or

King) from a standard deck of playing cards is C(52,12).

T or F

15. P(n,r) is equal to or greater than C(n,r) when n => 1.

T or F

16. There are 152 positive integers not exceeding 729 that are

divisible by either 7 or 13.

T or F

17. The relation An = an-1+ bn-2+ n + 2 is a linear,

homogeneous relation of degree 2.

T or F

18. A brand of shirt comes in three basic colors, has male, female

and unisex versions and has five sizes for each. This brand

has a maximum of 11 different varieties.

T or F

PART B

Each question is worth 6 points. Divided questions are worth 3

points for each section – or as indicated.

SHOW ALL WORK (within reason) in intermediate stages. Clearly

identify the final answer.

1. Determine:

A). -62 MOD 6

B). -84 MOD 7

2. Determine the Base10 expansion of (5CE) Base16

3. Define if the each set of integers are mutually relatively

prime. Defend your conclusion.

A). {8, 53, 77}

B). {7, 15, 29, 37, 42, 53}

4. Find the prime factors of the value 66,066. Show the result

in proper exponential form.

5. Given:

A = 102

B = 357

Define by factoring:

A). gcd (A, B) show in exponential form

B). lcm (A, B) show in exponential form

6. Using the Euclidean Algorithm, determine:

GCD (1980, 1950).

7. Convert (1011 1001) Base2 to:

A). ( ) Base16

B). ( ) Base10

8. Given 3716BASE10. Determine the equivalent value in BASE5.

Hint: Use the Euclidean Algorithm

9. Define: (show intermediate work)

A. P(13,3) =

B. C(11,2) =

10. What is the coefficient of ( x^5 y^4 ) in the expansion

(5x – 3y)^9 ? You may leave the answer in a proper

intermediate form.

11. Each locker in a building is labeled with five upper-case

alpha characters followed by two Base 16 characters. What

is the maximum number of different locker numbers that can be

generated?

12. A group of five fair coins are flipped seven times. What is

the probability that each result has four heads in each flip?

13. f(n)= 2*f(n/2) – 5 when n is even and f(1) = -3.

a. What is the value of f(4)?

b. What is the value of f(8)?

14. How many positive integers not exceeding 7356 are divisible

by neither 9 nor 21?

15. Given |A| = |B| = |C| = 80, |A INT B| = 20,

|B INT C| = 30, |A INT B INT C| = 10, and

|A UNION B UNION C| = 160 elements.

|A INT C| = ?

16. List the next SIX terms of the lexicographic ordering of the

n-tuple 34682 where each digit is in the set {2,3,4,6,8}.

17. Which lottery presents the player with the best odds for

winning, (A or B)? Defend your answer.

A = C(39,5)

B = C(40,6)

18. Determine if the following zero-one matrix is:

(2 points each)

a. reflexive T or F | 1 1 1 |

b. symmetric T or F | 1 1 0 |

c. transitive T or F | 0 0 1 |

Defend your answers.

……………………………………………….

OPTIONAL QUESTION (2 points or no points)

DO ONE.

A Develop the Basis Step of the algorithm to determine the

number of terms (cardinality) of the union of n mutually

intersecting sets. Show your work.

For example, the cardinality of the union of three mutually

intersecting sets is C(3,1) + C(3,2) + C(3,3) = 3+3+1 = 7.

B. Determine the Base7 value of 3954Base11. Show your work!!

END.

502q3c13c