# 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 ∧ 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 |