PART A
For problems 1 and 2:
Let A be a 4x4 matrix composed of all 0s.
Let B be a 4x4 matrix composed of all 1s.
1. A NAND B = all 0s.
True or False
2. A XOR B = all 1s.
True or False
3. The first nine digits of a book's ISBN-10 identifier is 0-441-27206. The tenth digit (the check digit) is 5.
True or False
4. The binary search algorithm has a smaller "Big(O)" notation than the linear search algorithm and is, therefore, the more efficient algorithm.
True or False
5. Given:
Matrix A is of the form 5 x 4
Matrix B is of the form 4 x 2
Matrix C is of the form 2 x 5
B x C x A has the form 5 x 5.
True or False
6. A Tautology is always true except when null sets are involved.
True or False
7. Given: A = {0,4,6,9,10} and B = {0,6,7,9,12}.
The Cartesian Product of A and B contains 25 valid elements since {0,0}, the NULL SET, is included in the total.
True or False
8. The cardinality of the power set {NULL, a, {NULL}, {{NULL}}} is 3. Note: {NULL} indicates the null (empty) set.
True or False
9. log(BASE2)10 = 3.3219 and log(BASE2)100 = 6.6439.
True or False
10. The matrix MEET is functionally the same as a matrix AND.
True or False
11. Using Relations f and g, it is possible that that fog = gof in most cases.
True or False
12. Using Set Identities, we can deduce that:
NOT A AND B = A OR NOT B.
True or False
13. Log(Base10)100 = 2; log 100(Base2) = 9.96
True or False
14. To be invertible, the involved functions must map one-to-one and onto.
True or False
PART B
SHOW ALL WORK (within reason) in intermediate steps. Solutions without intermediate work will be graded as zero.
Clearly identify each answer.
1. Define A*B where:
A = | 3 -3 6 | B = | 6 1 |
| 0 4 2 | | 0 -5 |
| 0 3 |
2. Given A and B
A = | 1 0 1 | B = | 1 1 0 |
| 1 1 0 | | 0 1 1 |
| 0 1 0 | | 1 1 0 |
Determine:
A). A MEET B
B). A JOIN B
3. Given A = -2*x^3 - 5x where x = (-0.3, 0, 0.3, 1.4). Determine:
a. CEILING A
b. FLOOR A
4. Define the "Big O" function of:
F(x) = 4x*log(x^2 + 7) + 5*[(4 + x^5) * log(x^7)+ 12]
5. Define the value of:
a) SIGMA (2*i) where i has the range i = 0 to 3.
b) PI (k + 4) where k has the range k = 0 to 3.
6. Define the values in the double sigma expression:
SIGMA1 SIGMA2 (3*i*j) where SIGMA1 has the range of j= 0 to 3, and where SIGMA2 has the range of i= 1 to 3.
7. Let A = (a,b,c,e,f,g,k) and B = (a,b,c,e,h,i,j). Determine:
a) A INT B
b) B – A
c) A – B
8. Let g be a function from the set G = {1,2,3,...34,35,36). Let f be a function from the set F = {1,2,3,...34,35,36}. Set G and F contain 36 identical elements (a – z and 0 - 9). A partial representation of the G and F relationships are:
g(1) = 26, g(2) = 17, g(3) = 22, and
f(6) = 1, f(9) = 3, f(11) = 2.
Assume a 1:1 and onto relationship. Determine:
a. fog
b. gof
9. Given f(w) = 2, f(x) = 5, f(y) = 2, f(z) = 3. What is the inverse function of 5? That is, f-1(5) = ?
10. Assume that the Basis Step for the sum of the first n ODD Integers is n^2. Develop a table of at least six exs to show this assumption seems to be true. Hint: Follow the process used in previous ECRs.
11. Assume n is a very large value. List the following elements in ascending order:
{(n log n^2), 2^n, log 20, 120, log 10, (n^2 log n^2) }
12. Define the value of:
PI SIGMA 3*k*j where SIGMA has the range k =0 to 4, and where PI has the range j = 0 to 3.
A. What is the best order to form the product ABCD if A, B, C and D are matrices with dimensions 15 x 25, 25 x 30, 30 x 20, and 20 x 15, respectively. Show your work! Limit your answer to the four alternatives listed below:
a) [( A * B ) * C ] * D
b) (A * B ) * ( C * D )
c) A * [ B * (C * D ) ]
d) [( C * D) * A ] * B
B. Solve for Computer Time Used when:
Problem size: n = 10^2
Bit operations used: n^3 log n
Processor speed: 10^10 operations/sec