Computer Organization and Architecture

The assignment content covers various topics in computer organization and architecture, including floating point representation, binary numbers, boolean functions, and boolean algebra. It provides solutions to questions on IEEE-754 single precision format for representing floating points, signed magnitude, one's complement, and two's complement in a 5-bit word, as well as a logic diagram for activating the CSU main entrance door during specific times using basic logic gates.

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Computer Organization and Architecture
Name
Affiliate Institution

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Table of Contents
Question 1 a: Floating points representation................................................................................................3
Solution:..................................................................................................................................................3
Question 1 b: Binary Numbers....................................................................................................................3
Solution:..................................................................................................................................................3
Question 2 a: Boolean Functions.................................................................................................................4
Solution...................................................................................................................................................4
Question 2 b: Boolean Algebra....................................................................................................................5
Solution...................................................................................................................................................5
References...................................................................................................................................................6
Bibliography................................................................................................................................................6

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Question 1 a: Floating points representation
A Computer uses IEEE-754 single precision format to represent floating points. What value (in
decimal) the computer represents if the floating point is represented using the following
binary digits:
0 01111110 10100000000000000000000
Solution:
01111110=126
Then, 126-127=-1
=1.10100000000000000000000 x 2^-1
= 0.110100000000000000000000 x 2^-1 x 2 ^1
=0.110100000000000000000000=0.5+0.25+0.0625=0.8125 Null & Lobur (2015).
Question 1 b: Binary Numbers
Using a "word" of 5 bits, list all of the possible signed binary numbers and their decimal
equivalents that are represent able in: [3+3+3 = 9 marks]
i. Signed magnitude
ii. One's complement
iii. Two's complement
Solution:
According to Patterson & Hennessy, (2012) a word of 5 bits can be represented as:
Signed
Magnitude
One’s
Complement
Two’s
complement
Decimal
Equivalent
00000 00000 00000 0
00001 00001 00001 1
00010 00010 00010 2
00011 00011 00011 3
00100 00100 00100 4
00101 00101 00101 5
00110 00110 00110 6
00111 00111 00111 7
01000 01000 01000 8
01001 01001 01001 9
01010 01010 01010 10
01011 01011 01011 11
01100 01100 01100 12

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01101 01101 01101 13
01110 01110 01110 14
01111 01111 01111 15
10000 10000 10000 16
10001 10001 10001 17
10010 10010 10010 18
10011 10011 10011 19
10100 10100 10100 20
10101 10101 10101 21
10110 10110 10110 22
10111 10111 10111 23
11000 11000 11000 24
11001 11001 11001 25
11010 11010 11010 26
11011 11011 11011 27
11100 11100 11100 28
11101 11101 11101 29
11110 11110 11110 30
11111 11111 11111 31
Question 2 a: Boolean Functions
Write a Boolean function and construct a logic diagram of a circuit which use of basic logic
gates to activate CSU main entrance door during 9:00 am to 12:00 pm and after lunch
time during 1:00 pm - 4:00 pm. You need to use 24-hour clock timing when designing
this circuit. [9 marks]
Solution
Let Boolean Variable A represent the time between 0900hrs to 1200hrs, variable B represent
time between 1300hrs to 1600hrs and Z represent the output.
The CSU main entrance can only be activated when either of the variables is true.
Boolean function is represented in the table below;
A B Z (A˅B)
0 0 0
0 1 1
1 0 1
1 1 1

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According to Rajaraman & Radhakrishnan, (2008). The logical circuit can thus be drawn from
the above function;
Question 2 b: Boolean Algebra.
Using basic Boolean algebra identities for Boolean variables A, B and C, prove that X’Y + XYZ’
+ Y’ + XZ (Y+Y’) = 1. Please show all steps and mention the identities used. [6 marks]
Solution
Starting with the second and fourth components;
xyz’+xz(y+y’)
= x(yz' + z(y + y'))
= x(yz' + yz + y'z)
= x(y(z + z') + y'z)
= x(y + y'z)
= xy + xy'z.
Any boolean variable A can be written in the following manner:
A = A(B + B') = AB + AB' since (B + B') equals to 1 for any boolean variable B, so the first
expression is not altered by AND-ing it with such an expression.
Recalling the first and the third components, the expression yields:
x'y + xy + xy'z + y'
= y(x + x') + y'(xz + 1)
= y + y' = 1
This proves that X’Y + XYZ’ + Y’ + XZ (Y+Y’) = 1 Villiers, (2010).

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References
Null, L., & Lobur, J. (2015). The essentials of computer organization and architecture, fourth
edition. Burlington, MA: Jones & Bartlett Learning.
Patterson, D. A., Hennessy, J. L., & Hennessy, J. L. (2012). Computer organization and design: The
hardware/software interface. Waltham, MA: Morgan Kaufmann.
Rajaraman, V., & Radhakrishnan, T. (2008). An introduction to digital computer design. India, II:
Prentice-Hall of India.
Villiers, M. D. (2010). Boolean algebra at school, vol 1. Place of publication not identified: Lulu
Com.
Bibliography
Arnold, B. H. (2011). Logic and boolean algebra. Mineola, N.Y: Dover Publications.
Whitesitt, J. E. (2010). Boolean algebra and its applications.