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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.

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

• Category:- Math
• Reference No.:- M91433

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