Logic Function and Boolean Algebra Report

Questions: 1. Explain why modern machines consist of multiple levels of virtual machines. Why not just have two levels, the digital logic level and the high-language programming level? 2. Given a (very) tiny computer that has a word size of 6 bits, what are the smallest negative numbers and the largest positive numbers that this computer can represent in each of the following representations? a. Signed magnitudeb. One's complementc. Two's complement3. Convert the followings: a. AC1216 to binary b. 10710 to 8-bit 2s Complement c. ********012 to Hexadecimal d. ********2 (8-bit 2s complement representation) to decimal 4. A Computer uses IEEE-754 format to represent floating points. What value (in decimal) the computer represents if the floating point is represented using the following binary digits: 110000010101000000000000000000005. Charles Sturt University (CSU) academic director want to hire a lecture in Sydney Study Center. The applicant has worked in three universities before; University of N ew South Wales (UNSW), Macquarie University, and Sydney University. Before he will hire him, he must get input from these three universities. After few weeks of receiving feedback from all three, he has come to the following conclusions: Hire if two of them say yes and one says no. Hire if all of them say yes. Don't hire otherwise.Construct a truth table and find the minimized Boolean function to implement the logic telling the director when to hire. Draw a circuit diagram for the Boolean function.6. Complete the truth table for the following sequential circuit: Answers: 1. In virtual system the original thing is not used. Only the image of the original this is used which looks like original to the user. There are so many places where the virtual environment is being used like Server, network, storage etc. There are so many techniques now a days which are being used by IT firms e.g. virtual tools, virtual machines etc. By using Virtual machines, IT firms create their virtual data centre which can be managed centrally by all branches of the IT firms in very low of budget. The virtualization can be done of so many things like server, storage devices; CPU etc. the most commonly used types of virtualization are- Process level, System Level and OS virtualization. Applications become larger and become complicated by using different types of technologies. Quality and usability attributes cannot be measured in the physical environment. That is why, virtual architecture helps to increase the quality and usability attributes. There may be many changes at all stages in the physical environment which is not easy to change at all places. Virtual environment helps in resolving these types of issues. Virtual environment is very beneficial in the time essential fields. For example when the time is very less and very critical, the virtual environment is very beneficial in those cases. It is very beneficial from the business point of view. E.g. in the banking system, the virtual takes few minutes setup while the traditional physical environment takes hours in setup. So the Virtual environment saves the time and money both. Virtual environment is very cost effective also. It saves the cost of the physical setup. Virtual environment minimizes the geographical limitation. The business can be easily connected with the consumers on the internet throughout the world. There is no problem of the location. The consumer can access the system from anywhere in the world. The consumer does not have to go outside from the home to buy the products. The competitive environment of any organization for virtual systems can be analysed by PEST and SWOT analysis very efficiently. The SWOT analysis is helpful to understand the market position of the company and helpful to implement future direction and the new proposed system. While the PEST analysis is used to evaluate external factors. The SWOT analysis measures the business unit and PEST analysis measures the trends and changes in the market. The SWET analysis analyse the following terms- Strength, Weakness, Opportunities and Threats. While the PEST analysis analyse the external factors which can affect the project like- Political Factors, Economic Factors, Social Factors and Technological Factors. The levels of modern virtual are more than digital logic and high language programming level; to secure the IT environment. These things are implemented by the modern virtual machines- Partitioning Hardware Independence Encapsulation Isolation If all the above four things get implemented in the IT environment, the disaster recovery would be very easily without disturbing any other unrelated component and the transaction will run without any trouble. The disaster recovery is very costly in the earlier environment or earlier virtual machines but in modern virtual machines which contain more levels, there are very easy, time saving and low budget disaster recovery techniques. Earlier it took 40 hours approximately while in modern virtual environment, it takes 4 hours approximately. Therefore is helps in saving time and resource utilization for the current systems. It is the demand of today and is helping to the IT firms in a very huge manner. In physical environment there were so many steps had to be followed like Configure Hardware, Install OS, Configure OS, Install backup, Start automatic recovery etc. But in modern virtual machines, there is no need to follow up all the above steps. It is only because of more levels than digital logic and high language programming level. (Microsoft 2014) 2 a) Largest Positive and Smallest Negative numbers in Signed Magnitude- Largest Positive: 011111(31) Smallest Negative: 100000(31) 2 b) Largest Positive and Smallest Negative numbers in ones complement- Largest Positive: 0111112(31) Smallest Negative: 1000002(-31) Ones complement works as a negative number of any numbers in arithmetic. It is obtained by inverting the binary numbers bits. 2 c) Largest Positive and Smallest Negative numbers in twos complement- Largest Positive: 0111112(31) Smallest Negative: 1000002(-32) (Digilent Inc. 2014) Twos complement works as a mathematical operation on binary numbers and it represents the binary signed number also. It is widely used in the computer language that is why it is most important. 3 a) Convert AC1216to binary Convert AC12 hexadecimal to Binary To convert hexadecimal number into binary number transforms each number into binary form. 1010 1100 0001 0010 Hexadecimal is a positional number system which with base 16. It contains and uses 16 numbers; 0 to 9 and A to F. A, B, C, D, E, F is used for 11, 12, 13, 14, 15 and 16 respectively. 3 b) 10710to 8-bit 2s Complement First convert 107 to binary because the binary representation of -107 would be equal to the 2s complement of 107- -107 binary representation= 2s complement of 107 Binary representation of 107 107/2 = (53) + 1 53/2 = (26) + 1 26/2 = (13) + 0 13/2 = (6) + 1 6/2 = (3) + 0 3/2 = (1) + 1 = (0) + 1 =1101011 The binary number gets obtained from the decimal number by dividing the number by 2 and then collecting the remainder numbers. Ones complement of 1101011= 0010100 Twos complement of 1101011= 0010101 Ones complement works as a negative number of any numbers in arithmetic. It is obtained by inverting the binary numbers bits and Twos complement works as a mathematical operation on binary numbers and it represents the binary signed number also. It is widely used in the computer language that is why it is most important. Twos complement can be obtained by adding 1 digit in ones complement. 3 c) 11001101012to Hexadecimal Make the pairs of 4 digits for the octal representation =0011 0011 0101 =335 3 d) 100111102(8-bit 2s complement representation) to decimal 10011110 to Decimal 1*(2)p7+0*(2)p6+0*(2)p5+1*(2)p4+1*(2)p3+1*(2)p2 +1*(2)p1+0*(2)p0 =1*128+0*64+0*32+1*16+1*8+1*4+1*2+0*1 =128+0+0+16+8+4+2+0 =158 (Stack Exchange Inc. 2012) Binary number gets converted into decimal number by multiplying each binary digit by 2 raise to power, starting from 0 from the right hand side and then by adding all numbers. 4. A Computer uses IEEE-754 format to represent floating points. What value (in decimal) the computer represents if the floating point is represented using the following binary digits: 11000001010100000000000000000000 To convert 11000001010100000000000000000000 to decimal in the computer; which uses IEEE-754; following steps have to be followed- Step1 First split the binary number 11000001010100000000000000000000 into 1 + 8 + 23 bits- 1 10000010 10100000000000000000000 The sign bit is 1, therefore the number is negative. Another 8 bits 10000010 are exponent And another 23 bits 10100000000000000000000 are mantissa Step 2- Add the hidden bit to the mantissa- 1.10100000000000000000000 Step 3- Convert exponent (10000010) to decimal and then subtract 127 = 130 -127 =3 2**3 Step 4- Now renormalize the mantissa- = 1.10100000000000000000000 times 2**3 = (1+ 1*(2**-1)+1*(2**-3) ) times 2**3 = (1+0.5+0.125) times 2**3 =1.625 times 2**3 =13 As the sign is negative, the 11000001010100000000000000000000 in decimal would be -13.0. (S.Orley, J. Methews 2000) 5. Charles Sturt University (CSU)academic director want to hire a lecture in Sydney Study Center. The applicant has worked in three universities before; University of New South Wales (UNSW), Macquarie University, and Sydney University. Before he will hire him, he must get input from these three universities. After few weeks of receiving feedback from all three, he has come to the following conclusions: Hire if two of them say yes and one says no. Hire if all of them say yes. Don't hire otherwise. Construct a truth table and find the minimized Boolean function to implement the logic telling the director when to hire. Draw a circuit diagram for the Boolean function. A, B and C is the input value and X is the result based upon the input conditions. If any of the two inputs are positive, the output result will also be positive and if all the three inputs are positive, the result will be positive but if any two of the inputs are negative the result will be negative. The candidate will be selected for the post of lecturer only if the result is positive; if at least two inputs are yes. The truth table is a mathematical table which is used in Boolean algebra logics and functions. It helps to understand typical logics easily. It helps to understand gates logics. Following truth table is describing the criteria to hire the applicant- A B C X 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 ABC=1 0 1 1 1 ABC=1 1 0 1 1 ABC=1 1 1 1 1 ABC=1 Therefore the output in the form of Boolean function would be- Output= ABC + ABC + ABC + ABC But the function is not minimized; following is the minimized function- = ABC + ABC + ABC + ABC = BC (A+A) + ABC + ABC =BC (1) + ABC + ABC =BC + ABC + ABC =B (C+ AC) + ABC =B (C + A) + ABC =BC + BA + ABC =BC + A (B + BC) =BC + A (B + C) =BC + AB + AC =AB + BC + AC (R.M. Dansereau, n.d) Now this is the minimized Boolean function for the truth table to appoint the applicant. Circuit diagram for the above minimized function is as follows- 6. Truth Table The truth table is a mathematical table which is used in Boolean algebra logics and functions. It helps to understand typical logics easily. It helps to understand gates logics. It shows the relationship between input and output of logic gates. Input of logic gates get displayed in left side columns and output get displayed on right side columns. The minimal Boolean function can be found with the help of truth table. Following example shows the truth table- A, B and C is the input value and X is the result based upon the input conditions. If any of the two inputs are positive, the output result will also be positive and if all the three inputs are positive, the result will be positive but if any two of the inputs are negative the result will be negative. A B C X 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 Following is the truth table for the given circuit- Next State A B X A B 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 References [1] Stack Exchange Inc. 2012, Mathematics, Viewed on 28th Jan 2015 https://math.stackexchange.com/questions/44866/conversion-from-binary-fraction-to-octal [2] R.M. Dansereau (n.d), Chapter III Boolean Algebra, Viewed on 28th Jan 2015 https://users.ece.gatech.edu/dva/ece2030/files/lec3.pdf [3] Microsoft 2014, Virtual Machine Architecture, Viewed on 28th Jan 2015 https://technet.microsoft.com/en-us/library/cc708365(v=ws.10).aspx [4] S.Orley, J. Methews 2000, A Quick Summary of IEEE 754 Notation, Viewed on 28th Jan 2015 https://class.ece.iastate.edu/arun/Cpre305/ieee754/homepage.html [5] Digilent Inc. 2014, Negative Binary Numbers, Viewed on 28th Jan 2015 https://learn.digilentinc.com/Documents/280