School of Mathematics * Service Teaching * Foundation Studies* 0N1: MATH19861

MATH19861: Mathematics for Foundation Studies (0N1)

Examination feedback January 2014:

1     (a) The distinction between closed and open intervals was clear to most students but many thought that intervals contain only integers; some students also tried to list the elements.
    (b) Generally correctly done with unnecessarily long justifications.
    (c) The concept of an universal set was unclear to many students.
2     Question 2a:    Most people correctly answered that the set of all tautologies of propositional logic is infinite. Some said it was finite because every tautology is fundamentally equivalent to T. While this is true, it's not what the question asked for! (If it had, it would have said something like "modulo fundamental equivalence".)
    Reasons given were generally along the right lines but most tried to explain in words how to construct infinitely many - I was looking for an actual construction. For example, give one "base" tautology to start (eg. p V ~p) and then extend it (eg. p V ~p V ~p V ...). Those that had slight mistakes/missing technicalities in their constructions or explained the process in words (but not explicitly) were docked a mark.
    Question 2b:     The majority got 2/4 on this question. Most identified the set was finite, but claimed it was the empty set. There is actually an integer divisible by every prime - zero! I gave full marks only if you said B = { 0 } or an equivalent statement.
    There were also a notable amount of people that gave a single counterexample for their reason and then said "so the set is finite". If you start with infinitely many elements in a set and prove that one element isn't in it, there are still infinitely many other elements in the set.
    Some people misunderstood the question as being whether there were finitely many integers divisble by A SINGLE prime and said that the set was therefore infinite. This isn't what was asked.
    Question 2c:     There are finitely many cats in Britain and the vast majority got this correct. Just because we don't know exactly how many there are, it doesn't mean there are infinitely many! There was a wide selection of colourful reasons and I essentially gave 2/2 to anyone who took the question seriously. It was a difficult question to give a reason for, but most people came up with something sensible.
    Finally, one general remark: some people in parts a) and c) attempted to say "it could be finite or infinite depending on your point of view" and give potential reasons for both without actually choosing one. I gave no marks for this, since it wasn't clear which answer was chosen. If you do any more maths exams in the future, you should always be very clear about your answer.
3 This question was answered correctly by a large number of students. It should be reiterated that it is useful to write the rows of the truth table in the `standard' order - see page 39 of the lecture notes. Unfortunately, a number of students incorrectly believed that F T, while some others did not realise that a truth table involving n variables should have 2^n rows. Students should also remember to check their work - this includes checking that s/he has copied the correct question.
4 Many candidates were successful in proving the tautology, although only one in five who attempted this question found a short proof. No points were given to those who reached the desiered conclusion "T", but made serious errors in the process. Two typical errors: incorrect application of the DeMorgan Law when rearranging ∼(∼p~(~p∨(q∨r)) and writing mixture of conjunctions and disjunctions without brackets, some thing like p∨qr ... and then incorrectly rearranging the expression.
5 On the whole, the question was well done with the vast majority of students achieving between 6 and 10 marks. However, there were a few common mistakes:
    1. Misinterpreting the order of the quantifiers. For example, one of the statements they were given went something along the lines of "there exists an x, for all y, p(x,y)". Some students interpreted this as "give me any y and I can find an x such that p(x,y) is true", which is not correct.
    2. Counter-examples to "there exists" statements. Some students would give one specific counter-example, which is enough to refute a "for all" statement but not enough for "there exists". They need to show why it can NEVER hold, not just one particular case.
    3. Proof of "for all" statements. Some students would show why the statement held at the end points of the given interval and that's all. They need an extra line to explain why that is enough to show that the statement also holds for the points in between.
6 Most people who did attempt it did well, many getting full marks. Nearly everyone got the base case correct and most people set up the induction well and had a clear understanding of the idea behind the proof. The calculation using the induction hypothesis was where most marks were lost, in particular a common error was assuming the statement was true for k and after a lengthy calculation showing it was true for k instead of k+1.
7 Most of the students were able to detect that (x+1) and (x-1) are factors of the proposed polynomial checking that -1 and 1 are roots. However less people are able to show that (x+1)^2 and (x-1)^2 are not factors of the proposed family. People whose showed this fact, use long division to show their arguments mostly. However a good amount of students use the criteria which relates the first derivative of the polynomial its roots and the roots of the polynomial. Calculation mistakes are present in long division.
8 A relatively easy question. Many students made unnecessary manipulations, but were not penalised if the conclusion was correct.


Course materials:
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Table of Symbols

When sending emails related to the course, you may wish to copy and paste mathematical symbols from the following table.

intersection  ∩
union  ∪
subset  ⊆
not a subset  ⊄
proper subset  ⊂
not a proper subset
 ⊄
element  ∈
not an element  ∉
emptyset  ∅
 
equivalent  ≡
negation  
for every  ∀
exists  ∃
conjunction "and"  ∧
disjunction "or"  ∨
conditional  →
biconditional  ↔
less or equal  
greater than equal  

Past exam papers: some warnings

Be aware that the syllabus may have changed, making some questions no longer relevant to the course you are taking. The number of questions may also have changed.

In January 2012 the examination paper will consist of 10 questions, each worth 10 marks. Any six of these questions should be answered. The mark will be out of sixty and will count 60 percent of the credit for this unit. A mark of 60/60 for the exam plus 100 percent in coursework will allow 100 percent to be scored for the course unit.

In this particular course unit, no calculators or formula tables can be used in tutorial classes or in the examination.

Email Policy:

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    Email: borovik >>at<< manchester.ac.uk
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