Mathematics in IT


Mathematics in IT

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Complete the following questions 1-14:

  1. Simplify the following Boolean expression: AB(A + B)(C + C)

 

  1. Design the combinatorial circuit for: (p’ *r) + q

 

  1. Design the combinatorial circuit for:

 

  1. Complete the truth table of the following Boolean expression:

 

  1. Prove or disprove that the following 2 expressions are equivalent. Use either the related laws for your proof, or complete the two truth tables:
    • (A + C)(AD + AD’) + AC + C
    • A + C

 

  1. The following is a message coded in ASCII using one byte per character and then represented in hexadecimal notation. What is the message?
    • 4469736372657465204D617468656D617469637320697320434F4F4C21

 

  1. Convert each of the following signed binary byte representations to its equivalent base-ten representation. What would each byte represent in Binary Coded Decimal? Show your work step by step.
    • 00010001
    • 01011100
    • 1111010

What is the signed binary sum of 1011100 and 1110101 in decimal? Show all of your work.

  1. Convert each of the following base-ten representations to its equivalent two’s complement in 7 bits. Show all of your work.
    • 12
    • -2
    • -8
    • 0

Define the highest and lowest integer that can be represented in this 7-bit two’s complement representation.

  1. What bit patterns are represented by the following hexadecimal notations? Show all of your work.
    • 9A88
    • 4AF6
    • DA

What is the hexadecimal sum of 9A88 and 4AF6 in hexadecimal and decimal? Show all of your work for all problems

  1. Consider the following graph:
    • Complete this table by finding the degree of each vertex, and identify whether it is even or odd:
Vertex Degree Even/Odd
A    
B    
C    
D    
E    
F    
G    
H    
    1. What is the order of the graph?
    2. Construct the 10 x 10 adjacency matrix for the graph.
  1. The graph below illustrates a switching network. The weights represent the delay times, in nanoseconds, travelled by a data packet between destinations, represented by the vertices.
    1. Complete the following table by finding the shortest distance and the path for that distance from vertex A to the other vertices:
Vertex Shortest Distance from A Path from A
B    
C    
D    
E    
F    
G    
H    
I    
    1. What is the shortest distance between A and J and the path for that distance?
  1. The following graph represents a portion of the subway system of a city. The vertices on the graph correspond to subway stations, and the edges correspond to the rails. Your job is to write a program for a cleaning car to efficiently clean this portion of the subway system.
    1. Using Euler’s theorem, explain why it is possible to pass through all of the stations by traversing every rail only once.
    2. Using Fleury’s algorithm, provide an optimal path to clean all the rails by passing through them only once.
    3. Is it possible to find an optimal path described in question 3-b that starts on any station? Explain your answer.
    4. Is it possible to find an optimal path described in question 3-b that starts and ends at the same station? Explain why or why not.
  1. A network engineer lives in City A, and his job is to inspect his company’s servers in various cities. The graph below shows the cost (in U.S. dollars) of travelling between each city that he has to visit.
    • Find a Hamiltonian path in the graph.
    • Find a Hamiltonian circuit that will allow the engineer to inspect all of the servers. How much will the cost be for his trips?
    • Is there another Hamiltonian circuit that will allow the engineer to inspect all of the servers other than your answer in question 4-b? If so, calculate the cost.
  1. Consider the following binary tree:
    • What is the height of the tree?
    • What is the height of vertex H?
    • Write the preorder traversal representation of the tree.
    • Write the array representation of the tree by completing the following table:

 

Vertex Left Child Right Child
A    
B    
C    
D    
E    
G    
H    
I    
J    
K    
L    
M    
N    
O    
P    
Q    
R    
S    
T    

 

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