Posted: February 18th, 2024

Mat1101 Assignment 2

Mat1101 Assignment 2
Trimester 1, 2024

Weight:45%Totalmarks:100 Due date: 23rd February 2024 11:59pm AEST
Course objectives relating to this assignment.
• Recognize and understand how numeric and character data is stored in a computer.
• Interpret and write simple algorithms in pseudo-code.
• Effectively use symbolic logic, to implement mathematical reasoning and construct proofs.
• Effectively communicate discrete mathematical concepts and arguments using appropriate mathematical notation.

Numeric and Character Data Storage
Computers store numeric and character data using binary digits, also known as bits. Each bit represents either a 0 or 1. Numeric data such as integers are stored by representing the number in binary form, with as many bits as needed to store the number. For example, the decimal number 10 would be stored as 00001010 in binary using 4 bits. Character data such as letters, punctuation, and other symbols are assigned numeric codes and stored as sequences of bits corresponding to the code. For example, the letter ‘A’ may be assigned the code 65 and stored as 01000001 in binary using 8 bits (Schneider, 2022).
Algorithms in Pseudo-Code
An algorithm is a set of steps to solve a problem or complete a task. Pseudo-code uses plain English words and syntax to describe the steps without specifics of a particular programming language. For example, here is a simple algorithm in pseudo-code to calculate the average of three test scores:
Define variables to store the three test scores and the average
Read the first test score and assign it to the first variable
Read the second test score and assign it to the second variable
Read the third test score and assign it to the third variable
Calculate the sum of the three test scores
Divide the sum by 3 to calculate the average
Output the average
Symbolic Logic and Mathematical Proofs
Symbolic logic uses symbols and formal syntax to represent logical concepts and reasoning. Propositional logic uses variables to represent statements that can be true or false, and logical operators like ¬ (not), ∧ (and), ∨ (or) to connect statements. Predicate logic builds on this by adding quantifiers like ∀ (for all) and ∃ (there exists) to make statements about properties of objects. This formal system can be used to implement mathematical reasoning and construct proofs. For example, a proof that the sum of two even numbers is even could be represented symbolically as:
∀x∀y[(Ex ∧ Ey) → E(x + y)]
Where E represents “is even”.
Communicating Mathematical Concepts
Appropriate mathematical notation is key to effectively communicating discrete mathematical concepts and arguments. Concepts are defined using precise notation and terminology. Statements are precisely formatted, with clear delineation between premises and conclusions. Logical statements and quantifiers are represented symbolically as in predicate logic. Diagrams, graphs, and visual representations are used where helpful for clarification. Proofs are structured with a clear logical flow indicated by punctuation. References to definitions, theorems, and previous results are precisely cited. Together, this allows for unambiguous communication of mathematical ideas, reasoning, and arguments (Krantz, 2007).
Submission instructions
The assignment is to be electronically submitted via Study Desk. If you cannot submit electronically, please contact the Course Coordinator as soon as possible to make alternative arrangements.
You are to submit your assignment as a Portable Document Format (PDF/A) file. PDF/A is an archival format that embeds all font glyphs used in the document within the PDF file. This means PDF/A documents display correctly on any computer system.
Word files will not be accepted by the assignment system. Instructions on how to save a Word document in PDF/A format are included on page 4.
Handwritten and scanned assignments are perfectly acceptable, as long as they
are submitted as a single (1) PDF file and follow the mathematical conventions used in the course. You just need to ensure that the resulting scanned assignment is clearly legible. Also ensure that the scanned pages are in the correct order and orientation. Some guidelines on scanning your handwritten assignment is included on page 6.
You must ensure regardless of whether your assignment is hand written or typed, that all mathematical notation etc. follow the mathematical conventions used in this course. A quick guide to presenting Mathematics correct in Word can be found on the Study Desk (if you really need to typeset your assignment). A guide to technical communications is given in Appendix B of the Study Book (also available as a separate document), both of which are available on the Study Desk.
If you have trouble submitting your assignment via the Study Desk etc., please contact the Course Coordinator, via UniSQAssist, email or phone ASAP.
Late submission of assignments
Students are expected to submit Assessment Items by the published due date. Should circumstances prevent a Student from submitting by the published due date, the Student may apply for an extension. Extensions may prevent feedback from being received in time to be used in preparation for subsequent Assessment Items.
Extensions sought during the Study Period will not be granted after the Assessment Item’s published due date — except where a Student can provide documentary evidence in accordance with the Assessment of Special Circumstances Procedure, that it was not possible to submit a request prior to the published due date.
Short extension
The purpose of a short extension is to provide a three (3) Calendar Day extension to accommodate unexpected, short-term interruptions that impact the Student’s ability to submit Assessment Items by the published due date. A short extension is only available once per Assessment Item and is applied to the published due date of the Assessment Item.
Applying for a short extension
All applications must be submitted via the Student Centre by the Assessment Item’s published due date. Applicants will receive approval of a short extension upon correct submission of the online form. The due date of the Assessment Item is extended by three (3) Calendar Days only, irrespective of whether the new due date is a University Business Day.
Long extension
The purpose of a long extension is to accommodate unexpected circumstances outside the Student’s control and which have a significant impact on the Student’s ability to meet the Assessment Item’s published due date. This may include factors such as a delayed Student placement or delays in receiving textbooks or learning materials. Other examples of Special Circumstances are listed in the Assessment of Special Circumstances Procedure.
A situation would not be treated as Special Circumstances when the circumstances are within the control of the Student or are to be expected in the normal course of the Student’s study, work, family and social life. Examples of such circumstances are indicated in the Assessment of Special Circumstances Procedure.
A long extension request must be supported by relevant and acceptable documentary evidence. Examples of acceptable evidence are listed in the Assessment of Special Circumstances Procedure.
If a Student receives a short extension and then experiences Special Circumstances that provide grounds for a long extension for the same Assessment Item, they may apply for a long extension of up to seven (7) Calendar Days from the revised due date.
Applying for a long extension
All applications for long extensions must be submitted via the Student Centre. These applications must be submitted prior to the revised due date of the Assessment Item, except where a Student can provide evidence in accordance with the Assessment of Special Circumstances Procedure that it was not possible to submit a request prior to the revised due date.
Normally, the maximum extension period is seven (7) Calendar Days however, where the Student can provide evidence in accordance with the Assessment of Special Circumstances Procedure that a longer extension is required, the relevant Associate Head (Learning, Teaching and Student Success) may choose to extend beyond seven (7) Calendar Days.
Academic Integrity
Your time at University is the time to develop the knowledge and skills you will need in future careers; and academic integrity is essential in ensuring the quality of your education and development. It celebrates the genuine achievement of yourself and your peers. Breaches of academic integrity (e.g. plagiarism, cheating, and collusion) undermine your development and the ability of you to operate at the level needed post graduation.
To help you understand academic integrity, you are required to undertake mandatory training once per year. This is available as a StudyDesk course. It will take approximately 30 minutes to complete and you are required to complete it once to apply to all the courses you are studying in 2024. You will be unable to access the link to submit your assessments until you complete the training.
We understand that when you take on a university study program, you might be in a time poor, high pressure situation. From time to time the temptation might be there to take shortcuts that lead to instances of academic misconduct.
If you are ever feeling pressured and the temptation to submit work that is not your own arises, don’t do it. Instead, use the free support services the University provides via:
• Your Course Course Coordinator or your Discipline Learning Advisor.
• Your Student Relationship Officer (SRO) for support and recommendations about your studies via appointment, phoning 1800 008 252 (Freecall within Australia), +61 7 4631 2285 or emailing UniSQ.support@UniSQ.edu.au.
If you require personal support and assistance, you can engage in UniSQ’s free, confidential and professional counselling services.
Student responsibilities
The assessment procedure also outlines the following student responsibilities:
• Students are responsible for submitting the correct assignment.
• If requested, Students must be capable of providing proof of date of submission of a assignment.
• If requested, students must be capable of providing a copy of assignments submitted. Copies should be despatched to the University within 24 hours of receipt of a request being made.
• Assignment submissions must contain evidence of student effort to address the requirements of the assignment. In the absence of evidence of student effort to address the requirements of the assignment, no mark will be recorded for that assessment item.
Steps required to produce a PDF/A file from Microsoft Word under Windows.
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 . In this dialogue window make sure the Save as type is PDF, as shown in Figure 1.
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 1: Save As dialogue with the Save as type: PDF circled. Note: Dialogue box will be slightly different in different versions of Word.

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 versions of Word.
e) Click on the Options button and a new window (Figure 3) will appear.
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.

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 versions of Word.
Scanning handwritten assignments
Some recommendations for scanning a handwritten assignment.
• Use a black pen as it will scan better.
• Scan at 150 – 200 dpi.
• Depending on your scanner you may need to either combine your images into a single document using Word, Preview (Mac OS X) or the free PDF Toolkit.
• Check the size of your document, the Maximum upload size is: 50 MB. If you exceed this try scanning at a lower resolution.
• View your PDF file in Adobe Reader (free download from http://www.adob e.com) to ensure that all symbols etc. have been correctly rendered, and the final document is readable.
Assignment instructions
• Show full working for each question. Give the marker every opportunity to understand how you obtained your answers. Your mathematical reasoning is just as important as the final answer. Part marks for each question will be awarded based on your mathematical reasoning.
• If no working is shown only part marks may be awarded for a correct final answer.
• Clear communication and good presentation (see page 7) will make it easier for the marker to give you marks for each question. Marks will be awarded for clear mathematical and technical communication. Tips on mathematical communication, see the UniSQ Library, Study Support – “Maths QuickTips” pages on mathematical and technical communication; and Appendix B Technical Communication of the Study Book. The “Maths Quick Tip” on Typing mathematics in MS Word gives hints on how to efficiently type mathematics.
• Generally, in all your calculations, use as many decimal places as your calculator will allow. Only round your final answers.
• Finally, your solution presented must only use “methods and techniques? discussed and used within the course. Marks will not be awarded for methods and techniques outside the scope of the course.
Marking Criteria
Communication and technical communication [10 marks]
A total of 10 marks will be awarded for demonstrating clear and consistent mathematical communication in the assignment. Students must demonstrate communication skills across two communication criteria outlined below.
a) Presentation and Communication [5 marks]
• Assignments must be legible and presented in the correct logical order. Pages are the in the correct orientation etc.
• All questions and parts clearly labelled. If a question is skipped, it should be clearly labelled.
• Assignments with missing sections, pages orientated incorrectly or in the wrong order will be penalized.
• Graphs/Tables are titled, and their point is clear.
• Sentences are used to provide an explanation of how you obtained the answer.
b) Mathematical Notation [5 marks]
• Well presented mathematics should be universally readable and apply accepted conventions. This includes for example alignment of equals signs in a sequence of calculations written down the page.
• Solutions that are approximations should be noted by the use of an approximation symbol, ˜. the course. For example, use of (*) for multiplication is not acceptable?, ?
• Assignments should use mathematical notation and conventions used in
mathematical notation. Also etc. should be used for logic expressions.
• Correct use of subscripts for variables and superscripts for powers, set notation etc. will be scrutinised where such notation is necessary to answer a topic question.
Level (Marks) Presentation and Communication Mathematical notation
Excellent (5)
Good with only minor issues (4)
Adequate but could be better (3)
Poor and needs work (2)
Well below expectations (1)
Almost non-existing/non-existent evidence (0)
Calculation Marking Guide [Total: 90 marks]
The total marks for each question are given at the start of each question. Each question is broken into a series of parts, the marks for each part of these questions are given in the left margin.
• Full marks for each part will be awarded to fully correct and well communicated answers.
• Marks will be deducted per calculation and other logic errors, up to the amount given to each part.
• Marks may be deducted if the marker cannot follow your reasoning or argument.
• Part marks may be given to correct answers without any reasoning or working.
Chapters covered: Study Book Chapters 1, 2, 3, and 4 Question 1: Leap year algorithm [10 marks] Consider the following algorithm.???????? ????????? ????????4 = 0???????? 100 = 0400 = 0??
1. Input a four digit year.
2. if ( mod ) { Divisible by 4 }
2.1 if ( mod ) { Divisible by 100 }
2.1.1 if ( mod ) {Divisible by 400}
2.1.1.1 isLeapYear TRUE;
2.1.2 Else
2.1.2.1 isLeapYear FALSE;
2.2 Else { Not divisible by 100}? 2.1.3 End if
2.2.1 isLeapYear TRUE;
3. Else { Not divisible by 4 }? 2.3 End if
3.1 isLeapYear FALSE;
4. End if
marks 5. Output isLeapYear 2564
4 a) Trace the algorithm starting with the input .
b) Document the pseudo-code that you would need to add to the algorithm after step 4 to output the number of days in February of the input year. Make sure that you describe the changes in detail and add a comment to each line of 6 marks pseudo-code.

Question 2 [45 marks]
Everything stored on a computer is expressed as a sequence of bits (0s and 1s). However, different types of data (for example, characters and numbers) may be represented by the same sequence of bits. Hence, depending on requirements computers can be custom designed for specific roles. For example, a simple computer controller does not need the same precision as a super computer used for weather calculation. Hence, the number of bits used to store numbers can be significantly different. In this question, we will consider a 12-bit computer controller based on the following specifications.
Specifications of the 12-bit computer controller
Text characters (or symbols) are represented using 8-bits in our computer. the hexadecimal value representing the state of these 8-bitsTable 1 maps the “Control, Basic and Supplemental Latin 1 Character set” to68a. For example, this hexadecimal value to binary gives the state of the 8-bits (i.e. the 8-bits0100 0100 from Table 1 the character ‘h’ has the hexadecimal value . Converting
represents the character ‘h’).
In this computer, we will also assume that numbers (both signed/unsigned integers, and floating point numbers) are stored in 12-bits. Floating point (real) numbers are stored as per the algorithm in the Grossman (2009), with as the exponent bias (where ?? is the number of bits for the characteristic).2??-1 – 1
6-bits of these 12-bits reserved for the mantissa (or significand) and
For example, the string of 24-bits:0011 0110 0011 1001 0011 0101, in this computer might represent:× 3×
Table 1); or 2 • three ASCII/LATIN-1 characters ‘695’ (i.e. 8-bits encoded as per
• two numbers ( 12-bits). The interpretation of these 12-bits will be
different depending whether the numbers are stored as:– signed integers (e.g. 867 and -1739), or0.13672 -0.00040440.13672 ).
– as floating point (real) numbers (e.g.0.1406More precisely, any floating point number between-0.0004044 -0.0004119 and and
will have the same 12-bit pattern, in this not very accurate computer. Similarly, any floating point number between and will also have the same 12-bit pattern.
a This mapping is based on Unicode 10.0 Standard
Table 1: Hexadecimal map of the “Control, Basic and Supplemental Latin 1 Character set” to an 8-bit encoding scheme. See http://www.unicode.org/charts/PDF/U0000.pdf and http: //www.unicode.org/charts/PDF/U0080.pdf for control character definitions.
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) What is the largest positive floating point (or real) number that is represent5 marks able using the 12-bits on this computer.
b) Find the value of the 12-bits required to represent the signed integer:-15
3 marks on this computer.
c) Find the value of the 12-bits required to represent the floating point number10.01
5 marks on this computer.
d) Is the number stored in Question 2(c) exact? If not, what is the actual number
1 mark stored?
e) Find the actual bit pattern required to store the word below.
Apple .
3 marks
The remaining parts of Question 2(0101 0011 1111 1101 0111 1101f—i) refer to the following 24-bits:
3 marks f) Represent these 24-bits as a hexadecimal number.
3 marks g) What characters according to Table 1 are represented by these 24-bits?
4 marks h) What pair of signed integers is represented by these 24-bits?
6 marks i) What pair of floating point numbers could be represented by these 24-bits?
j) This computer controller also supports 24-bit (i.e., double precision ) floating point (real) numbers using the method outlined in Grossman (2009), except in this case of double precision floats all 24-bits are used to store a single number, with 8-bits of these 24-bits being used to store the characteristic. Using this information answer the following:
i) What is the smallest positive floating point (real) number that can be10.01 6 marks represented using double precision on this computer?
ii) What will be the state of the 24-bits, if is stored as a double preci6 marks sion floating point number on this computer? Is it exact?
Question 3 [17 marks] ?? ??
A complex farm machine is controlled by two sensors and . Each sensor only has two states 0 and 1. The following logic rule controls if the machine runs:(?? ? ??) ? (¬?? ? ??) ? (¬?? ? ¬??) . 1
(1)
That is, the machine will run when the above logic rule (Equation 1) returns (i.e., True).
a) Construct a truth table for each of the following logic expressions.?? ? ??¬?? ? ??¬?? ? ¬??
4 marks i)
4 marks ii)
4 marks iii)
b) Combining the results from Question 3a create a truth table for the rule in
Equation 1 which controls the farm machine. Hence, determine the state of 5 marks the sensors when the farm machine is running.
Question 4 [18 marks]
In computers, colours are created by blending different combinations of red, green html, photoshop, gimp etc. These 6 digits represent the state of the 24-bits. For??62929 These colours are normally specified as three two-digit hexadecimal numbers in example, Brown is specified as to indicate the proportions of red, green and
and blue (RGB). The RGB combination required to represent a colour on a computeris stored in 24-bits. These 24-bits are divided inshade of red, green or blue. As 8-bits are used for each colour, colours can store 256shades of red, green or blue. Hence, some 16 million colours (represented on must modern computers in a single image.3 × 8-bits, which store a specific224 or 2563) can be
blue required. Hence, the bit pattern:1010 0110 0010 1001 0010 1001,
will be interpreted as “Brown”. For grey shades the three proportions will always000000 ????????????????????0000001111 1111 1111 1111 1111 11110000 0000 0000 0000 0000 00001111 1111 0000 0000 0000 00000000 0000 1111 1111 0000 00000000 0000 0000 0000 1111 1111.????000000???? ;;;, be equal. Moreover indicates that the colour is fully saturated. Hence, white corresponds to or the bit pattern: Black which is represented by the bits: fully saturated red is or the bit pattern: fully saturated green is which is represented by the 24-bits: and fully saturated blue is which has the bit pattern:
a) Convert the RGB values for the colours below to their equivalent 24-bit pat-
Colour name Colour Hexadecimal???? 8?? 8????2 22 22 terns.
Rosy Brown
4 marks Firebrick
b) Convert the 24-bits representing the colours below to their equivalent hexa-
Colour name Colour 24-1111 1111 0001 0100 1001 00110011 0010 1100 1101 0011 0010bit Pattern decimal values.
Deep Pink
4 marks Lime Green
each pixel is stored in 24-bits. These can be either stored in 2row or column× 3 c) On computers, images are broken up in several million pixels. The colour for
order. For example, consider the image below consisting of pixels, with the hexadecimal representation of each pixels 24-bit pattern shown.
FFA07A FFD700 FFA07A
FFA07A FFA07A FFA07A
The 144-bits required to store the image can be stored in bits as: FFA07A FFD700 FFA07A FFA07A FFA07A FFA07A (row order), or
FFA07A FFA07A FFD700 FFA07A FFA07A FFA07A (column order). Hence, to recreate an image you need to know the total number of pixels, the storage3 × 3 order and one other dimension of the image.
i) The following 216 bits, store a pixel image in column order.
00FF00 00FF00 00FF00 FFFFFF 00FF00 FFFFFF FFFFFF 00FF00 FFFFFF.
6 marks Sketch the image represented by this bit pattern. 7680 × 4320
ii) How many bits are required to store an 8k UHD image ( ) image in 24-bit RGB colours? How many whole 8k images can you store 4 marks in an 8GB (GigaByte) drive.
References
Grossman, P. (2009), Discrete Mathematics for Computing, 3rd edn, Palgrave MacMillan.
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