Question 1 (24 marks)
Table 1: Hexadecimal map of the “Control, Basic and Supplemental Latin 1 Character set” to an 8-bit encoding
scheme, as per Unicode 10.
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 NULL SOH STX ETX EOT ENQ ACK BELL BS HT LF VT FF CR SO SI
1 DLE DC1 DC2 DC3 DC4 NAK SYN ETB CAN EM SUB ESC FS GS RS US
2 SP ! ” # $ % & ‘ ( ) * + , – . /
3 0 1 2 3 4 5 6 7 8 9 : ; < = > ?
4 @ A B C D E F G H I J K L M N O
5 P Q R S T U V W X Y Z [ \ ] ^ _
6 ` a b c d e f g h i j k l m n o
7 p q r s t u v w x y z { | } ~ DEL
8 XXX XXX BPH NBH IND NEL SSA ESA HTS HTJ VTS PLD PLU RI SS2 SS3
9 DCS PU1 PU2 STS CCH MW SPA EPA SOS XXX SCI CSI ST OSC PM APC
A NBSP ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ SH ® ¯
B ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿
C À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï
D Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß
E à á â ã ä å æ ç è é ê ë ì í î ï
F ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ
a) Using the Unicode mapping given in Table 1 complete the following.
4 marks i) Write down the hexadecimal representation of 48-bits required to store the
text string below.

4 marks ii) Find the characters represented by the bit pattern below.
0101 0111 0101 1011 1010 1101
b) Suppose we have a computer in which every integer must be stored in 8-bits with
the same number of negative and non-negative integers, find:
4 marks i) the integer represented by the 8-bits: 1111 1111.
4 marks ii) the range of signed integers that can be represented.
c) In a computer where every real number must be represented in 10-bits, of which
four (4) are assigned to the characteristic, find, assuming that the bias for a ?-bit
characteristic is 2
?−1 − 1:
4 marks i) the 10-bit floating point representation for −100.
4 marks ii) the real number represented by the following 10-bits: 0 1001 10001.
Question 2 (24 marks)
6 marks a) Complete the truth table (Table 2).
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Table 2: Truth Table
? ? ¬? ¬? ? ∨ ¬? ¬? ∨ ¬? (? ∨ ¬?) ∧ (¬? ∨ ¬?)
T
T
F
F
6 marks b) Using the truth table (Table 2) or otherwise, deduce a simpler expression (i.e. logically equivalent expression) for the proposition.
(? ∨ ¬?) ∧ (¬? ∨ ¬?).
Justify, why your answer is logically equivalent and correct.
c) A digital circuit takes three logical inputs ?, ?, ? and produces the logical output
? (?, ?, ?) as set out in Table 3.
Table 3: Logical output of ? (?, ?, ?), for values of the logical input variables ?, ? and ?.
? 1 1 1 1 0 0 0 0
? 1 1 0 0 1 1 0 0
? 1 0 1 0 1 0 1 0
? (?, ?, ?) 1 0 1 0 1 0 0 0
6 marks i) Write down the disjunctive no

Question 3 (28 marks)
a) The following function ? (?) is defined recursively as:
? (?) =
⎧
⎨
⎩
5? (? − 1) − 6? (? − 2), ? ≥ 2
0, ? = 0
1, ? = 1
4 marks i) Compute ? (2) and ? (3).
6 marks ii) Solve the recurrence. That is, find a general non-recursive formula for the
recurrence.
6 marks iii) Find the particular solution which satisfies the initial conditions ? (0) = 0 and
? (1) = 1.
4 marks b) Write down the boolean expression which corresponds to the following logic circuit.
?
?
Legend:
Not And Or
c) Consider the universal set ? = {?, ?, ?, ?, ?, ? , ?, ℎ, ?, ?}, and the sets ? = {?, ?, ?, ?},
? = {?, ?, ?}, and ? = {?, ?, ?, ?}.
4 marks i) Find the Cartesian product of ? × ?.
2 marks ii) Find the set defined by ? ∪ ?.
2 marks iii) Find the set defined by (? ∪ ?) − ?.
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Question 4 (24 marks)
a) Consider the graph below.
?1
?2 ?3
?4
?6 ?5
?7 ?8
?9
?10
6 marks i) Write down the adjacency matrix for this graph.
6 marks ii) Analysing the adjacency matrix, determine if the graph is Eulerian, SemiEulerian or neither. Justify your answer in terms of the relevant theory. If
the graph is Eulerian or Semi-Eulerian write down a Eulerian path through
the graph.
6 marks b) Consider the following weight matrix.
? =
⎡
⎢
⎢
⎢
⎢
⎣
? ? ? ? ?
? 0 26 ∞ 17 ∞
? 26 0 12 ∞ 15
? ∞ 12 0 14 8
? 17 ∞ 14 0 13
? ∞ 15 8 13 0
⎤
⎥
⎥
⎥
⎥
⎦
(2)
Draw the weighted graph represented by the above weight matrix.
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6 marks c) Consider the following graph (Not to scale).
B A
C D
E
40
22
12
16
14
18
4
Compute and draw the minimal spanning tree for the above graph.
Question 5 (20 marks)
During this course you have been introduced to mathematical theory that applies to various aspects of modern computing. Below are some extensions to these topics and ideas
covered in this course. Write a page or two on each of them. Your discussion should cover
the points listed and include four to five references, other than the text book2
.
Topic 1: Computer Representation of Images.
Your discussion should cover:-
• The difference between RGB and CMYK colour models. Also where they are most
commonly used in computing.
• The differences between RGB and RGBA.
• The differences between 8-bit, 16-bit, 24-bit and 32-bit colour images.
• Should also touch on the storage requirements for a “1920 × 1080” image under the
various colour models listed above.
Topic 2: Binary trees in computers
Your discussion should cover:-
• What is a Binary Tree? Illustrate the definition using an appropriate diagram.
• What is a Binary Search Tree? It should also explain how you construct a Binary
Search Tree? Illustrate this process for the numbers: {8, 3, 1, 10, 6, 14, 4, 7, 13}.
• How you can search for a number in a Binary Search Tree. Illustrate, your discussion using the Binary Search Tree created above.
• Sorting algorithms for Binary Search Trees. Again illustrate with reference to the
Binary Search Tree created above.
End of Assignment Questions
2
For this assignment use the Harvard referencing system,
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Steps required to produce a PDF/A file from Microsoft Word under Windows.3
a) Save the document as a .docx file.
b) Go to the File menu and select Save As.
c) You should now see the dialogue box similar to Figure 1
4
. In this dialogue window
make sure the Save as type is PDF, as shown in Figure 1.
Figure 1: Save As dialogue with the Save as type: PDF circled. Note: Dialogue box
will be slightly different in different version of Word.
d) Select More options… from the Save As dialogue box shown in Figure 1. A new
window outlining extended Save As options will appear as shown in Figure 2.
Figure 2: Word extend Save As PDF options, which allows various options for the PDF to
set. Note: Dialogue will be slightly different depending on version of Word.
e) Click on the Options button and a new window (Figure 3) will appear.
3 These instructions assume Word 2018 on Windows 10. All other versions will be similar but not
identical.
4 Depends on the version of Word
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Figure 3: Word extend Save As PDF options, which allows various options for the PDF to
set. Note: Dialogue will be slightly different depending on version of Word.
Make sure that ISO 19005-1 compliant (PDF/A) is selected as shown in Figure 3. Once completed click OK.
f) Save the file with an appropriate filename. If it is your final assignment submission
make sure you include your student number and course code in the file name.