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Examiners’ commentaries 2020 Examiners’ commentaries 2020 MT2116 Abstract mathematics Important note This commentary reflects the examination and assessment arrangements for this course in the academic year 2019–20. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE). Information about the subject guide and the Essential reading references Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2011). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If different editions of Essential reading are listed, please check the VLE for reading supplements – if none are available, please use the contents list and index of the new edition to find the relevant section. General remarks Learning outcomes At the end of this course and having completed the Essential reading and activities you should: have used basic mathematical concepts in discrete mathematics, algebra and real analysis to solve mathematical problems in this subject be able to use formal notation correctly and in connection with precise statements in English be able to demonstrate an understanding of the underlying principle of the subjects be able to solve unseen mathematical problems in discrete mathematics, algebra and real analysis be able to prove statements and formulate precise mathematical arguments. Showing your working We start by emphasising that you should always include your working. This means two things. First, you should not simply write down the answer in the examination script, but you should explain the method by which it is obtained. Secondly, you should include rough working (even if it is messy!). The examiners want you to get the right answers, of course, but it is more important that you prove you know what you are doing: that is what is really being examined. We also stress that if you have not completely solved a problem, you may still be awarded marks for a partial, incomplete, or slightly wrong, solution; but, if you have written down a wrong answer and nothing else, no marks can be awarded. So it is certainly in your interests to include all your workings. 1
MT2116 Abstract mathematics Knowing the definitions In this course, precision and clarity are extremely important. It is vital that you know the key definitions and theorems exactly, so that you can quote them and use them. It is simply not possible to prove something using a formal definition if you only have a vague and incorrect recollection of what that definition is. How could you use the formal definition of convergence of a sequence to prove that a sequence converges if you do not know what it is that you need to establish, because you do not know the definition? Well, you cannot. It is so important to know the definitions, and there are a number of marks to be picked up simply for knowing them. Covering the syllabus and choosing questions You should ensure that you have covered the bulk of the syllabus in order to perform well in the examination: it is bad practice to concentrate only on a small range of major topics in the expectation that there will be lots of marks obtainable for questions on these topics. The examination paper has some element of choice: your best six questions (out of eight) count. If you have not covered the whole syllabus, then you will be limiting your choice. Assuming you have, however, covered the whole syllabus, it is a very good idea to take a little time to choose carefully: it could be that a question on your favourite topic is in fact more difficult than a question on another topic. Expectations of the examination paper Every examination paper is different. You should not assume that your examination will be almost identical to the previous year’s: for instance, just because there was a question, or a part of a question, on a certain topic last year, you should not assume there will be one on the same topic this year. Each year, the examiners want to test that candidates can reason precisely mathematically, and that they know and understand a number of mathematical concepts and methods. In setting an examination paper, they try to test whether the candidate does indeed know the methods, understands them, and is able to use them, and not merely whether he or she vaguely remembers them. Because of this, every year there are some questions which are likely to seem unfamiliar, or different from previous years’ questions. You should expect to be surprised by some of the questions. Of course, you will only be examined on material in the syllabus, so all questions can be answered using the material of the course. Examination revision strategy Many candidates are disappointed to find that their examination performance is poorer than they expected. This may be due to a number of reasons, but one particular failing is ‘question spotting’, that is, confining your examination preparation to a few questions and/or topics which have come up in past papers for the course. This can have serious consequences. We recognise that candidates might not cover all topics in the syllabus in the same depth, but you need to be aware that examiners are free to set questions on any aspect of the syllabus. This means that you need to study enough of the syllabus to enable you to answer the required number of examination questions. The syllabus can be found in the Course information sheet available on the VLE. You should read the syllabus carefully and ensure that you cover sufficient material in preparation for the examination. Examiners will vary the topics and questions from year to year and may well set questions that have not appeared in past papers. Examination papers may legitimately include 2
Examiners’ commentaries 2020 questions on any topic in the syllabus. So, although past papers can be helpful during your revision, you cannot assume that topics or specific questions that have come up in past examinations will occur again. If you rely on a question-spotting strategy, it is likely you will find yourself in difficulties when you sit the examination. We strongly advise you not to adopt this strategy. 3
MT2116 Abstract mathematics Examiners’ commentaries 2020 MT2116 Abstract mathematics Important note This commentary reflects the examination and assessment arrangements for this course in the academic year 2019–20. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE). Information about the subject guide and the Essential reading references Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2011). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If different editions of Essential reading are listed, please check the VLE for reading supplements – if none are available, please use the contents list and index of the new edition to find the relevant section. Comments on specific questions Candidates should answer SIX of the following questions: THREE from Section A and THREE from Section B. If additional questions are answered, only your best THREE answers from Section A and your best THREE answers from Section B will count towards the final mark. All questions carry equal marks. Section A Answer any three questions from this section. Question 1 (a) In this part of the question, x denotes a real number and S is the following statement. S : if x is rational then x 2 is rational. i. Show that S is true, by giving a proof. ii. Write down the contrapositive of S. Is it true, and if so why? iii. Write down the converse of S. Is it true? If so, prove it. If not, write down a counterexample. (b) Let p, q and r be statements. Using a truth table or otherwise, determine whether the following two statements are logically equivalent: S1 : (p =⇒ r) ∧ (p =⇒ q) S2 : (p =⇒ (r ∧ q)). 4