University Assignment: Data Representation and Boolean Algebra

This assignment solution addresses Boolean algebra concepts, data representation, and digital logic. The first part converts a decimal number into the single-precision IEEE 754 format and provides binary conversions for a 5-bit word using signed magnitude, one's complement, and two's complement. The second part represents clock hours in a 5-bit binary form, creates a timing diagram, and develops a logic diagram using basic logic gates to control a door based on the time. The solution also includes a proof for a Boolean expression and a comprehensive bibliography of relevant sources.

Running head: BOOLEAN ALGEBRA
Boolean algebra
Name of the student
Name of the University
Student Id
Subject code (ITC544)
Assignment 1: Data representation and digital logic
Author note

Page 1 of 7BOOLEAN ALGEBRA
Answer 1:
a)
Single precision IEEE 754 format- 0 01111110 10100000000000000000000
In this section, the first MSB (most significant bit) is the sign, the middle portion is the
exponent and the last portion is the mantissa.
Decimal conversion- 8.125 * 10^-1
b)
5-bit word- 0 1010
Signed magnitude- +10 (Most significant bit-0 [positive] rest 4 bits are magnitude)
One’s complement- 10101 (all the 0’s are replaced by 1 and vice-versa)
Two’s complement- 10110 (1 is added to the one’s complement)
Answer 2:
a)
This section will represent the hours of the clock in 5-bit binary form.
Magnitude Binary Clock pulse (p)
1 A B C D E High (+1)
00001
2 00010
3 00011
4 00100
5 00101
6 00110
7 00111
8 01000

Page 2 of 7BOOLEAN ALGEBRA
9 01001
10 01010
11 01011
12 01100
13 01101 Low (0)
14 01110
15 01111
16 10000
17 10001
18 10010
19 10011
20 10100
21 10101
22 10110
23 10111
24 11000
The timing diagram will supposedly use a 12-hour clock period. For 1-12 time on the
clock, the pulse will give +one value. For 13-24, the pulse will give zero value. To address the
requirements, the 12 am in the clock is considered as one, which will continue until the pulse
reaches 24. The door will open for 9 am (9) to 12 pm (12) and 1 pm (13) to 4 pm (16). In
addition, the binary bits will be represented by A B C D E to represent the required bits in the
logic diagram. The circuit will be made by the use of basic logic gates.
The logic diagram is:
A'BC'D'EP+A'BC'DE'P+A'BC'DEP+A'BCD'E'P=Q (For clock= +1)
A'BCD'EP’+A'BCDE'P’+A'BCDEP’+AB'C'D'E'=R (For clock=0)

Page 3 of 7BOOLEAN ALGEBRA
Fig: Logic diagram
(Source: Created by the author)
b)
X’Y+XYZ’+Y’+XZ (Y+Y’)
= X’Y+XYZ’+Y’+XZY+XZY’
= X’Y+XY (Z+Z’) +Y’+XZY’
=X’Y+XY+Y’+XZY’

Page 4 of 7BOOLEAN ALGEBRA
=Y+Y’+XZY’
=1+XZY’
=1 (PROVED)

Page 5 of 7BOOLEAN ALGEBRA
Bibliography:
Aswal, A., Perumal, G., & Prasanna, G. S. P. (2012). On basic financial decimal operations on
binary machines. IEEE Transactions on Computers, 61(8), 1084-1096.
Braun, E. L. (2014). Digital computer design: logic, circuitry, and synthesis. Academic Press.
Brown, F. M. (2012). Boolean reasoning: the logic of Boolean equations. Springer Science &
Business Media.
Brzozowski, J. A., & Seger, C. J. H. (2012). Asynchronous circuits. Springer Science & Business
Media.
Clote, P., & Kranakis, E. (2013). Boolean functions and computation models. Springer Science
& Business Media.
Givant, S., & Halmos, P. R. (2012). Lectures on Boolean algebras. Springer Science & Business
Media.
Hammer, P. L., & Rudeanu, S. (2012). Boolean methods in operations research and related
areas (Vol. 7). Springer Science & Business Media.
Harris, K., & Montalbán, A. (2014). Boolean algebra approximations. Transactions of the
American Mathematical Society, 366(10), 5223-5256.
Kumar, N., Gupta, P., Sahu, M., & Rizvi, M. A. (2013, March). Boolean Algebra based effective
and efficient asymmetric key cryptography algorithm: BAC algorithm. In Automation,
Computing, Communication, Control and Compressed Sensing (iMac4s), 2013
International Multi-Conference on (pp. 250-254). IEEE.
Lidl, R., & Pilz, G. (2012). Applied abstract algebra. Springer Science & Business Media.
Luke, Y. L. (2014). Integrals of Bessel functions. Courier Corporation.
Mano, M. M. (2017). Digital logic and computer design. Pearson Education India.

Page 6 of 7BOOLEAN ALGEBRA
Moisil, G. C. (2014). The algebraic theory of switching circuits (Vol. 41). Elsevier.
Novák, V., Perfilieva, I., & Mockor, J. (2012). Mathematical principles of fuzzy logic (Vol. 517).
Springer Science & Business Media.
Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis (Vol. 12). Springer Science
& Business Media.
Trypuz, R., & Kulicki, P. (2013). On deontic action logics based on Boolean algebra. Journal of
Logic and Computation, 25(5), 1241-1260.