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MAST20026 Assignment
Creation date：2024-04-29 15:06:44
Belonging category：数学mathematics
Assignment

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MAST20026 Assignment 1

This assignment consists of 2 pages.
Assignments are due in Gradescope on the due date listed above. Please see the instructions on the page where
you accessed this file for details on submitting your assignment. To ease submission, please prepare assignment
solutions for each question on its own page. To enable to graders to grade anonymously, do not include your
name anywhere on your assignment.
Students are encouraged to work together on understanding the problems and their solutions. However, submitted
solutions must be prepared individually, in your own words, and without the aid of others. For example, if you
are asked to give an example as part of your solution, your example should be meaningfully different from anyone
you have worked with on understanding the problems. Answers presented without justification will receive no
marks.
If you do not completely understand your submitted solutions, it is possible you are committing academic
misconduct. This is a serious offense.
Please do not upload this assignment paper or any of its questions to any online “help” service (e.g., CourseHero,
Chegg, etc...). By doing so, you are actively ruining the learning experience for yourself and future students and
more than likely violating the University’s academic misconduct policy.
PART A
To goal of assessments in this course is to help you know how well you are mastering the concepts presented in
lecture and tutorials. The work that you hand in for assessments is meant to reflect your level of understanding.
And so it is important that the work you hand in fully demonstrates your mastery of subject concepts.
Download the file called Guidelines for Good Mathematical Writing from the same place you accessed this file.
Read it and write a short response (≤ 250 words) that considers the following questions:
• Was there anything about the reading that you found surprising?
• What other experiences, if any, do you have in mathematics classes where you were expected to write in
full sentences as part of your solutions? Do you find it difficult to write in mathematics courses?
• How will your approach to crafting your submission for this assignment change, if at all, as a result of
having done this reading?
Note: In the reading above, the author uses the terms formal and informal differently than how we are using
them in our subject materials.
5 marks
PART B
(2) (a) (4 marks) For each of the implications below, write the contrapositive and the converse.
(i) If n = 41, then n is composite.
(ii) If x′ > 2r − 4, then −2x′ − 2 < r.
(b) (4 marks) Let p(x) : x > 2 over the domain R.
(i) If one exists, give an example of a condition q(x) for which p(x) =⇒ q(x) is true for every
x ∈ R, but q(x) =⇒ p(x) is not true for every x ∈ R. If no such example exists, briefly
explain why not.
(ii) If one exists, give an example of a condition q(x) for which p(x) =⇒ q(x) is true for every
x ∈ R and q(x) =⇒ p(x) is true for every x ∈ R. If no such example exists, briefly explain
why not.
1
(3) Consider the following theorem and its partially completed proof.
Theorem. There are infinitely many prime numbers.
Proof.
i. Let p be the largest prime number. (premise)
ii. Let p1, p2, . . . , pk be the list of prime numbers that are less than p. (i)
iii. Let n = p1p2 · · · pkp+ 1. (notation)
iv. n > p (iii)
v. n is not prime. ( )
vi. None of p1, p2, . . . , pk, p divide n. (iii, algebra)
vii. There exists a prime q ∈ [p+ 1, n− 1] that is a divisor of n. ( )
viii. q > p. (vii)
ix. p is not the largest prime number. (viii)
x. ( ). (i,ix)

(a) (1 marks) Fill in the blank on line x.
(b) (4 marks) What justification(s) should go in the blanks for on lines v. and vii.?
(c) (3 marks) Using the steps above, write out the proof as a informal proof. Feel welcome to use any
of the sentences above in your proof.
(4) (3+3 marks) Re-write the proof of Theorem 1.24 from the Week 2 notes as both a formal proof and as
an informal proof.
(5) Our goal in this exercise is a first attempt at making precise our intuition surrounding limits at infinity.
(a) (1 mark) Give an informal one sentence explanation of your understanding of the statement
The limit of 2x+ 1 as x goes to ∞ is ∞.
(b) (3 marks) For x, r ∈ R, let p(x, r) be the condition
2x+ 1 > r
Find a value of k so that p(x, 10) is true for every x > k. Justify your response.
(c) (1 mark) Find a value of k so that p(x, 31) is true for every x > k. Justify your response.
(d) (1 mark) Find a value of k so that p(x, 71) is true for every x > k. Justify your response.
(e) (0 marks) Find a value of k so that p(x, 10001) is true for every x > k. Justify your response.
(f) (2 marks) Find a formula for k as a function of r so that p(x, r) is true for every x > k. Justify your
response.
(g) (1 mark) Informally, your formula in the previous part allows to verify the following fact: for any
r ∈ R, f(x) = 2x+1 eventually gets bigger and stays bigger than r. Consider the following definition:
Definition. Let f : R → R be a function. We say the limit of f(x) as x goes to ∞ is ∞ when for
every r ∈ R there exists k ∈ R so that f(x) > r for every x > k.
Are you convinced the limit of f(x) = 2x + 1 as x goes to ∞ is ∞, as defined by the definition
above? Explain your thinking with a sentence or two by referring to your answer from (f) and the
definition in (g). (It is okay if the answer is “No, I am very confused by this question”.)
(h) (3 marks) Given the definition of the limit of f(x) as x goes to ∞ is ∞ given above, what do you
expect the definition of the limit of f(x) as x goes to ∞ is −∞ to be? Give your answer both as a
sentence and as a statement using the language of formal logic.
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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