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COMP9020 Assignment
创建日期：2024-06-11 17:58:56
标签： COMP9020
Assignment

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COMP9020 Assignment 

Submission is through inspera. Your assignment will be automatically submitted at the above
due date. If you manually submit before this time, you can reopen your submission and
continue until the deadline.
If you need to make a submission after the deadline. Unless you are granted
Special Consideration, a lateness penalty of 5% of raw mark per 24 hours or part thereof for a
maximum of 5 days will apply. You can request an extension up to 5 days after the deadline.
Answers are expected to be provided either:
• In the text box provided using plain text, including unicode characters and/or the built-in
formula editor (diagrams can be drawn using the built-in drawing tool); or
• as a pdf (e.g. using LATEX) – each question should be submitted on its own pdf, with at
most one pdf per question.
Handwritten solutions will be accepted if unavoidable, but we don’t recommend this approach
as the assessments are designed to familiarise you with typesetting mathematics in preparation
for the final exam and for future courses.
Objectives and Outcomes
The purpose of this assignment is to build your mathematical maturity in the areas of Number Theory
and Sets and Languages. Most questions are presented at an abstract level so that the consequences are
very general, and can be applied in a variety of situations: not just later on in the course, but also beyond.
The specific motivation for each problem can be summarised as follows:
Problem 1: This question works through a proof of Bézout’s Identity: the fact that the gcd of x and y can
be written as an integer linear combination of x and y. To do this we work through an approach
common throughout the course: Introducing a new definition based on known concepts; exploring
that definition with some examples; and then proving results about the definition.
Problem 2: This question asks you to prove several set theory identities using the formality of the laws
of set operations. Formal mathematical proofs are similar to programming because we are trying
to reach an end goal by following very specific rules. We will see closely related proofs again when
we cover Propositional Logic, and more generally, Boolean Algebras.
Problem 3: In this question we explore proofs involving formal languages. This question brings together
the concepts of Sets, Languages and Number Theory and results from all areas will be needed to
complete the proofs.
After completing this assignment, you will:
• Be able to make rigorous arguments about the foundational structures used in discrete
mathematics [All problems]
• Understand several deeper connections between different topics of the course [Problems
1 and 3]
1
Advice on how to do assignments
Collaboration is encouraged, but all submitted work must be done individually without consulting
someone else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism”
policies.
• Assignments are to be submitted in inspera.
• When giving answers to questions, we always would like you to prove/explain/motivate your
answers. You are being assessed on your understanding and ability.
• Be careful with giving multiple or alternative answers. If you give multiple answers, then we
will give you marks only for your worst answer, as this indicates how well you understood the
question.
• Some of the questions are very easy (with the help of external resources). You may make use
of external material provided it is properly referenced1 – however, answers that depend too
heavily on external resources may not receive full marks if you have not adequately demonstrated
ability/understanding.
• In the assignment specification, questions have been given an indicative difficulty level:
Pass Credit Distinction High distinction
This should be taken as a guide only. Partial marks are available in all questions, and achievable
by students of all abilities.
1Proper referencing means sufficient information for a marker to access the material. Results from the lectures or textbook
can be used without proof, but should still be referenced.
2
Specific advice on how to do this assignment
Problem 1
The main purpose of any assignment is to assess your ability and understanding. A simple way to
do this is to introduce you to a novel concept based on definitions you are familiar with; ask some
exploratory questions to get you familiar with the concept; and then ask you to prove some results
about the new concept. In this question we do this by defining a family of sets Sx,y (i.e. one set for each
pair of integers x, y); checking you are comfortable with the definition (parts (b) and (c)); and then
proving results about the set which ultimately lead to Bézout’s identity (parts (d) – (h)). The aim of
the question is to show that the gcd of two numbers x and y is the smallest positive number that can
be written in the form mx+ ny where m and n are integers (this is known as a linear combination of x
and y). We do this by taking d to be the gcd and z to be the smallest positive linear combination of x
and y and showing that d = z.
Important!
This question is asking you to prove Bézout’s identity based on the principles from lectures.
You should not make use of Bézout’s identity in the process of the proof!
• Part (a) introduces a result that you will use throughout the problem (and the course), and it
provides a gentle introduction to the process of providing rigorous proofs.
Important!
It is important that you prove this part (and parts (d) – (h)) in the abstract – that is, that
you keep things general and use x, y, z, m, and n rather than proving the result for specific
values. This will mean that the result can be applied in any situation.
When working with proofs, a good first approach is to unravel definitions until you have some-
thing you can manipulate, and then reapply definitions until you reach the result. As an example,
here is an HD-level solution for a related question:
Prove that for all x,m, z ∈ Z, if x|z then x|mz.
Suppose x|z.
Then z = kx for some k ∈ Z (by the definition of |).
So for any m ∈ Z: mz = m(kx) = (mk)x.
As k and m are integers, (mk) is an integer, so x|mz as required.
• Parts (b) and (c) are intended to get you familiar with the set Sx,y for two pairs of x, y – namely
4, 6 and −6, 15. The set S4,6 is the set
S4,6 = {4m+ 6n : m, n ∈ Z}
so elements of this set are found by having m and n take various integer values. To show 2 is the
smallest positive element of S4,6 you need to show two things:
– You need to show that 2 ∈ S4,6 – i.e. find integers m, n such that 2 = 4m+ 6n; and
– You need to prove that 1 /∈ S4,6. As this is fundamentally a “negated” statement, it could be
useful to prove this by contradiction – that is, assume that 1 ∈ S4,6 and derive a contradic-
tion. To use part (a), you need to find an integer z such that z|4 and z|6. This should tell you
something useful about elements of S4,6, in particular, why there is a problem if 1 ∈ S4,6.
3
• For part (c), it may be a good idea to try and find some elements of S−6,15 and see if you notice
a pattern. You do not need to include this experimentation in your final solution, unless you
believe it adds to your demonstration of ability and understanding. To justify your answer, you
should go through similar steps to part (b), but there may be additional steps required to exclude
possible smaller values.
• For parts (d) –(h) you should be working in the abstract - that is, you should be keeping things
general and using x, y, d and z rather than specific values – with the exception of the case
when z = 0. It is easiest to consider this case separately, noting that z = 0 automatically places
restrictions on x, y and d.
• To show a set A is a subset of B, you need to show that every element of A is also an element
of B. In the case of part (d), the set A is the set Sx,y and the set B is the set of integer multiples
of d. So you need to prove that number that can be written in the form mx+ ny where m, n ∈ Z
is multiple of d. Note that you have to prove this for all m and n, not specific values. Since we
are working with arbitrary values for x, y, and d, the only way to prove this is to work from the
definitions (just as with part (a)).
• To show that d ≤ z, we note that, by definition, z ∈ Sx,y. So what does the previous result now
tell us about z and d? Be careful – this doesn’t immediately give the result, you need one more
observation – namely that d and z are positive. You may use the result of (d), even if you haven’t
proven it, to answer this question.
• Once you have a proof that z|x, the proof that z|y will likely follow in a similar way (because x
and y are interchangeable in all definitions). It is acceptable to write “Similarly z|y”.
• The hint says to show that z|x you should consider x % z. If z|x, what does that mean for x % z?
Now consider the following two observations:
– 0 ≤ (x % z) < z (from the lectures)
– z is the smallest positive element of Sx,y
So if we could show that (x % z) ∈ Sx,y what does that mean? How might we show (x % z) ∈
Sx,y? Well, as we are working in the abstract, we cannot do much with x % z other than rewrite
it according to its definition.
• To show z ≤ d recall the definition of d and consider what we have shown about z in the previous
question. You may use the result of (f), even if you haven’t proven it, to answer this question.
• For part (h) we now have to show every multiple of d is an element of Sx,y. The hint tells us to
us (e) and (g) – this should give you one multiple of d in Sx,y. How might we be able to extend
that to show that every multiple of d is in Sx,y?
Problem 2
For this problem you are asked to prove the identities using the Laws of Set Operations (and any
derived laws proven in lectures).
Important
Full marks for these questions will only be awarded to proofs using the Laws of Set Operations.
Partial marks are available for proofs that do not use the laws.
Formal proofs, as we require here, are much stricter than other types of proofs encountered in this
course (or even this assignment!). There are two main reasons why we require formal proofs for these
questions:
4
– Formal proofs highlight the close connection between mathematical reasoning and programming
in Computer Science. When programming, you have to be precise with your instructions because
computers are very strict with their execution. If we can write mathematical arguments with the
same level of strictness, then this gives us a basis for having computers “reason”.
– We will see the same set of laws twice more in the course when we cover Propositional Logic
and Boolean Algebras. This means that proofs built solely from the Laws of Set Operations will
translate directly to those other topics – giving us multiple results for free.
Here is an example of a formal proof using the Laws of Set Operations:
Prove that ∅c = U :
∅c = ∅c ∪∅ (Identity of ∪)
= ∅ ∪∅c (Commutativity)
= U (Complementation)
• To ensure full marks, each step should be justified with a single rule. That rule may apply in
multiple, non-overlapping, places (not currently implemented in the Proof Assistant).
• You may use Uniqueness of Complement and/or Principle of Duality, and these may be useful
for part (d). These are not currently implemented in the Proof Assistant either.
• You may prove and use your own intermediate results if it helps (also not implemented in the
Proof Assistant).
Problem 3
This question brings together the topics of Sets, Languages and Number Theory. You are asked to
prove and disprove set-based identities, but because the constructed sets are based around formal
languages, the Laws of Set Operations do not apply. Instead, we will need to prove these results from
first principles, together with results from Number Theory.
• Part (a) is intended to familiarise you with the languages L2 and L3. The languages Σ=2 and Σ=3
have only finitely many words in them, so can be explicitly listed out. L2 and L3, on the other
hand, have infinitely many words so cannot be completely listed. Instead you should provide
a description of these languages, based on their construction rather than on properties such as
the one used in part (b). A good description makes it easy to check if a given word belongs to
the language or not. A few examples following the description can be helpful. As an example,
here is a reasonable informal description of the language XY where X = {a}∗ and Y = {a}∗:
XY consists of words of the form xy where x is a word of 0 or more a’s and y is a word
of 0 or more a’s. That is, XY = {λ, a, aa, aaa, . . .}.
Note that the observation that XY = {a}∗ was not part of the description, as this is something
that should be proven (see below)
• A counterexample to a "for all" type of proposition, is a single example which shows that the
proposition is false. In part (c) we want to show that
(Σ=2 ∩ Σ=3)∗ ̸= (Σ=2)∗ ∩ (Σ=3)∗.
To show two sets are unequal it is sufficient to find an element of one which is not an element of
the other (note that λ is an element of both sides!). You may use part (d) (if proven) if it helps.
5
• The only approach to proving set equality (covered in lectures) that applies in parts (d) and (e) is
to show A ⊆ B and B ⊆ A. Here is an example of such a proof, using the sets X and Y defined
earlier:
Prove that XY = {a}∗:
We first show that XY ⊆ {a}∗.
If w ∈ XY then w = xy where x ∈ X and y ∈ Y. As X = Y = {a}∗, we have x and y
are a sequence of 0 or more a’s. Therefore xy is a sequence of 0 or more a’s, and hence
xy ∈ {a}∗.
We now show that {a}∗ ∈ XY.
Let w ∈ {a}∗. Then w = wλ. As w ∈ {a}∗ = X and λ ∈ Y we have that w ∈ XY.
We have shown that XY ⊆ {a}∗ and {a}∗ ∈ XY, therefore XY = {a}∗.
• For part (d) it may help to review the definition of the least common multiple from the Number
Theory content.
• For part (e), another way of describing Σ∗ \ {a, b} which may be helpful is:
Σ∗ \ {a, b} = {λ} ∪ Σ≥2.
案例分类
标签云
关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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