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MAT 246 computational
创建日期：2024-05-20 16:55:32
所属分类：数学mathematics
标签： MAT 246
computational

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MAT 246

Please note:
1. Ideally the PS must be written in the spaces provided, and page by page (a picture or scan of individual pages) to be
uploaded to the appropriate pages of a Crowdmark file (link to be posted.) If you wish, you can write your answers on
separate pieces of paper and upload them properly at each page of the Crowdmark. Please note that any mismatch in
uploading the pages of the solutions may cause in parts of the solutions not getting marked.
2. Please provide your final, polished solutions in the spaces provided. If you have no access to a printer you can write your
solutions on a blank sheet of paper and upload a picture of scan. Remember, it is an art to write a short yet complete
solution; one learns a lot from practicing this art. Please write a complete, even though long solutions. Then inspect and
edit; while editing, ask yourselves:
- am I proving common believes and facts accepted in the literature? If yes, know which facts are common belief and
which ones need an argument.
- can I just quote some of the facts that I tried to prove? This requires a good familiarity with the literature of the
subject (in our case, the textbook and the slides, and earlier questions in the same PS).
- am I introducing too many cases in my proof, and presenting parallel arguments? Change your type of argument if
possible (direct proof versus proof by cases, or proof by contradiction.)
Editing your first complete solution is a very good way of learning. Make sure you have a good solution, then return to
it and reflect on the above questions.
3. Problem set questions contain ideas complementary to the textbook and lecture materials, opportunity for reflection
and deepening on the subject. The level of difficulty of the questions is elementary so that each student has chance to
individually think about the problems. So, please don’t let your “kind" friends take this opportunity away from you!
4. Not all questions will be marked, nor they are all of equal weight, and it is not known in advance which questions will be
marked, and how heavy they will be weighed. Therefore if a question is skipped one may lose a larger portion of the PS
mark than one would expect.
5. To receive full mark on the computational questions all the computation details must be provided. For questions involving
proofs there is no need to prove textbook or “slides" proofs but one can simply quote them. When a theorem is applied
the relevance of it to the proof must be made clear.
6. Collaboration in this work is not allowed. You may discuss the idea of a the solutions only, but each person must write
their own solutions completely independently, without knowing or consulting the structure and details of the argument of
another person.
So please be careful not to share any written hint with your friends, because some people have photographic memory,
and then your solutions may become subject to plagiarism.
7. By participating in this problem set you agree that you are aware of the university regulations regarding plagiarism.
Individual students are expected to organize and write their own solutions independently. During the online teaching,
and while students are connected through email, any assistance to your fellow classmate in the form of a written note
might be directly copied and it may count as a form of plagiarism. Markers are carefully monitoring the solutions, and
any incidence of plagiarism, and all the parties involved will be dealt with according to university regulations. Please
carefully read the item regarding plagiarism in our course outline, and consult the university policies regarding plagiarism
(not knowing the rules is not a good excuse for not obeying them):
http://www.artsci.utoronto.ca/newstudents/transition/academic/plagiarism
Page 1 of 3
MAT 246, PS2 first draft Due: Fri. Mar. 3 Time 10:00 pm. EST
-
On PS1, unfortunately over %10 of the class did not properly acknowledged the statement below. As
a result they lost their PS1 mark. Please note that an acknowledgement is when you sign your name at
the bottom of a statement, as a sign of agreement with what the statement is suggesting.
00 If I sign my name on top of a page, what am I acknowledging?
0. Failure to answer this question may result in a mark of 0 on the entire PS2. And it is not the name,
signature and date, but it is the acknowledgement .. signing some statement that is intended in this part:
Please sign below as an acknowledgment that you have read the cover page ; (unsigned papers shall not be marked.)
Print Name (first, last) : , Signature: Date:
1. (a) We like to solve 51x ≡ 1 (mod 100). Note that 100 is not prime, so we cannot automatically apply
cancellation law, as corollaries of chapter 4 guaranteed. Translate this equation to a Diophantine equation,
apply Euclidean Algorithm to solve it and determine a solution to the original equation.
(b) RSA: read the discussion of RSA on top of page 54, and consider N = 253 and the encrypter E = 9. With
this protocol, a person wants to send the message of M = 13, what would they actually send us (R)? And in
another occasion someone sends us the message 37. What was their intended message M?
2. a) Read Lemma 7.2.9 and its proof. This proof goes against our vision of transition between chapters because it
is using a proof technique from chapter 4! Use Lemma 0 (from the lecture slides week 5 or 6) to give a different
proof (a very short proof) of this lemma that uses gcd(s, u) = 1 (Hint: read lemma 7.2.2).
(b) prove if a and b are integers and m > 1, and a ≡ b (mod m) and that gcd(a,m) = 1, then gcd(b,m) = 1 as
well.
(c) Prove that if a is multiplicative inverse of b in mod m then both a and b are relatively prime to m.
3. Consider two natural numbers a and b that are relatively prime. Define the sets Sa and Sb to be the set of all
natural numbers (between 1 and a, including 1) that are relatively prime to a and b respectively. Similarly let
Sab that is the set of all numbers less than ab which are relatively prime to ab.
a) Prove that any number of the form ar + bs, where s ∈ Sa and r ∈ Sb must be relatively prime to ab.
b) Conclude from part (a) that φ(a)φ(b) ≤ φ(ab).
4. (a) Prove the following statement in two ways, by applying PMI and by applying PCMI: : with the quantifiers
over natural numbers, ∀n∃i∃j such that n = 2i−1(2j − 1), that is, any natural number can be written as the
product of powers 2 and the left over, which is an odd number.
(b) Prove for each n the choice of (i, j) must be unique.
Page 2 of 3
MAT 246, PS2 first draft Due: Fri. Mar. 3 Time 10:00 pm. EST
(c) How are the facts in (a) and (b) used to prove the function f : N×N −→ N defined by f(i, j) = 2i−1(2j − 2)
is a bijection between N × N and N? What is the interpretation of this bijection in terms of the infinite hotel
puzzle?
5. (a) Given that g : A×A −→ A is a bijection, show that the function G : A×A×A −→ A is also a bijection:
G(m,n, k) = g(g(m,n), k) for all (m,n, k) ∈ A×A×A
(b) Use the idea of part (a) and the previous question, and an argument using mathematical induction to prove for
each n ∈ N there must exist a bijection Fn
Fn : N× N× · · · × N −→ N
between the Cartesian product of number of n copies of N and N.
6. Greek Constructions, Golden ratio, Golden rectangle, Golden Triangle, Construction of 36◦:
This question involves reading ahead, some accessible material from chapter 12, and some Greek Constructions. (Please
consult Tutorial Week 6 exercise on Greek Construction and read the slides on Greek Construction, posted on the
LECTURES module.)
(a) Read theorem 12.2.15 and its proof. Replace the proof with a proof using Pythagorean identity applied to
triangle MDC. This proof is shorter and does not need theorems 11.3.9 and 11.3.11.
(b) Golden Ratio: For positive real numbers a > b the ratio ab is said to be Golden ratio, if
a
b =
a+b
a . A rectangle
with sides a and b is a golden rectangle. Denote the golden ratio ab by g, and prove that g must be a root of
the polynomial x2 − x− 1. Determine the value of g.
(c) Golden Rectangle: Start with a precisely drawn square ABEF (read the vertices counterclockwise with A
being the lower left vertex) with each side of length 2. Let O be the mid-point of AB. Show that the line
segment OE is
√
5, and use Greek Construction to extend the original square to a golden rectangle that you call
it ACDF (counterclockwise). Make sure your Greek Construction steps are precisely drawn and documented.
(d) Golden Triangle and Construction of 36◦: Consider the isosceles triangle with the angles 36◦ and 72◦ . Let the
base be of size b, and let the sides be of size a each. Prove that the ratio ab is a golden ratio (as in part (b)).
(Hint: see Theorem 12.4.10.) Use this information, and your Golden rectangle in part (c) to construct a angle of
36 at the vertex A. This consrtruction must be executed in three steps of Greek Construction. Demonstrate
your construction, and indicate your steps and explain which rule was used in each step.
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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