Mathematics in computing

Assignment Help Computer Engineering
Reference no: EM13897

Question 1

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.

2. A XOR B = all 1s.

3. The first nine digits of a book's ISBN-10 identifier is 0-441-27206. The tenth digit (the check digit) is 5.

4. The binary search algorithm has a smaller "Big(O)" notation than the linear search algorithm and is, therefore, the more efficient algorithm.

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.

6. A Tautology is always true except when null sets are involved.

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.

8. The cardinality of the power set {NULL, a, {NULL}, {{NULL}}} is 3. Note: {NULL} indicates the null (empty) set.

9. The matrix MEET is functionally the same as a matrix AND.

10. Using Relations f and g, it is possible that that fog = gof in most cases.

12. Using Set Identities, we can deduce that:

NOT A AND B = A OR NOT B.

13. Log(Base10)100 = 2; log 100(Base2) = 9.96

14. To be invertible, the involved functions must map one-to-one and onto.

PART B

SHOW ALL WORK (within reason) in intermediate steps. Solutions without intermediate work will be graded as zero.

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 examples 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

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