Office Hour: If you want to discuss something with me then send me an email and we can arrange a time (of course you can also do this by talking to me immediately after a lecture). You can also ask questions by email (as well as in person). If you turn up at my office and I am free and not in the middle of something that I need to push on with, then I will be happy to talk to you/answer your questions. We are supposed to set an office hour so here is mine: Thursdays, 1-2 - (but let me emphasise that I am available at many other times).

My room is Alan Turing Building 1.120

Internal tel. 55875

email mprest@manchester.ac.uk

MATH20302, Propositional Logic

Course Notes (complete) Typo: page 14, line 11, replace "left" by "right" - now (16th April) corrected, also an undefined notation (''v entails S'') (on page 28, lines -5, -4, -1, and page 29, lines 1, 3, 6) has been replaced by our standard one (''v(S)=T'').

An example of a (de)construction tree for a propositional term.

Truth table for the problem we looked at in lecture 2. Assume that the following statements are true: (i) everyone who doesn't take Logic takes Algebra or Mechanics; (ii) everyone who doesn't take Algebra takes Statistics and either Mechanics or Logic; (iii) everyone who takes Mechanics and Logic takes neither Algebra nor Statistics. Does it follow that (Q) no-one takes both Logic and Mechanics? We express each of the conditions (i), (ii) and (iii) as a propositional term in the propositional variables l,a,m,s and similarly with the last statement. One method of solving the problem is to construct the truth table, that is, consider all possible valuations on these four variables. The question then becomes: does every valuation that makes each of (i), (ii) and (iii) true make (Q) true? In terms of truth tables this reads: for every row where each of (i), (ii) and (iii) is True, is (Q) True?

Exercises 1

Solutions 1, Solution to Q5, Solution to Q6

Exercises 2

Solutions 2

Exercises 3

Solutions 3

Exercises 4

Solutions 4

Exercises 5

Solutions 5

Exercises 6

Solutions 6

Exercises 7 Changed 21st March: in Q3, 2.1.14 has been replaced by 2.1.15 (also changed in the solutions).

Solutions 7 (2.1.14 should read 2.1.15)

Exercises 8 and Solutions 8. This material (on a Gentzen/natural deduction-style calculus) is entirely optional (it won't be examined on this course but it is the type of calculus used in the 3rd year Predicate Logic course) but the contrast with the Hilbert-style calculus is interesting (there are not axioms, just rules of deduction, and it allows for much more natural, and more easily found, proofs).

Exercises 9

Solutions 9 and the separate Solution to Q1(b).

Here are the Examples Class Examples from Week 10 and the Answers to these.

and here are the Examples Class Examples from Week 11 (delete 1(e)(ii) which just repeats part (i)) and the Answers to these.

Courseworks There will be two take-home pieces of coursework, each weighted at 10 percent. You'll have about 10 days to do each. The second will be put up here on Monday 22nd April and will be due in by 4pm on Thursday 2nd May.

Coursework 1 and Solutions to Coursework 1 (including a few comments on common errors and now, 21st March, the marks for each question).

Coursework 2. Here are some notes on definable sets which explain some of the notation in Question 2(b) of the coursework. Answers to Coursework 2

Review Class Friday 17th May I will hold an extra review class from 4.30-(approx)6 on Friday 17th May in Renold J17.

Comments on, and comparison with, last year's exam The course this year will not differ significantly from last year. But the exam will be a bit harder than last year's since that proved to be too easy and I had to scale down marks at the top end quite strongly. By "harder" I mean that it will be more difficult to get very high marks. Here, for your information, are my comments on last year's (the 2011/12) exam. "Performance on the exam was generally very good. Question A5 (deciding whether or not an assertion is true and then proving it or giving a counterexample) was, predictably, the least well-done question in Section A. Question B9 (Compactness) was, also predictably, the least popular choice of question in Section B but the questions in that section were done better than I had expected. In retrospect, the exam was a bit too straightforward and I scaled down in the 1st and 2i range (100 stayed at 100 and 60 stayed at 60); even after scaling, the average mark was still very high, reflecting a very good performance by the great majority of students." I also gave some detailed comments on some of the questions on that exam, as follows: solutions/comments to questions A5, B7-9 on the 2011/12 exam (9th May - a 'typo' in the solution to Q8 now corrected).

Past exam papers You can find the exams from 2010, 2011 and 2012 on the University website. Here is the 2009 Exam. The 2011 Exam I've also posted here the Solutions to the 2011 Exam (they're a bit faint but I seem to have thrown away the originals, so can't rescan).

Comments on this year's exam

• Of past papers, 2012 is most directly relevant (that's when I took over the course);
• there are some pretty obvious minor notation and terminology changes between 2012 and before that;
• some questions before 2012 ask students to produce proofs within a formal calculus - you will not be asked to do that;
• there is more emphasis now on predicate logic and this topic will be tested on this year's exam (some previous exams don't have it).

Further comments on this year's exam

• To emphasise (since students still ask), I won't ask you to produce formal proofs in the Hilbert-style calculus in the exam.
• As regards proofs in the usual sense - I don't expect you to learn long proofs from the course. There are some proofs at the end of Part 2 (around the connections between different versions of Completeness and Compactness) that you should try to understand and know well enough that you could use them in an exam, namely, 2.1.11 (Finiteness of proofs), Statement of 2.1.12 (Completeness v1), 2.1.13 (Completeness v2), 2.1.14 and 2.1.15 (Compactness v1, v2). And you should try to understand how to use the Compactness Theorem (the last question in Section B is often this).
• It was pointed out to me that I didn't actually name the Disjunctive Normal Form Theorem in the printed notes (sometimes the exam asks for a statement of this). It is Proposition 1.4.5 (and the Conjunctive Normal Form Theorem is Proposition 1.4.7).

MATH19812 (0B2) (Foundation Year Mathematics, Semester 2)

For general information about the course (syllabus, arrangements, how the course is assessed, further reading) follow the link just above. That page also contains all the course notes and the relevant HELM notes as well as all the examples sheets. On this page and on blackboard I will put up, week-by-week, the examples sheets and the corresponding solution sheets. I will also use this page and blackboard for any further general information and, for example, solutions to courseworks tests.

Since I'm using mostly the visualiser for lectures, it might be useful to have scans of the resulting sheets available in case you missed anything. So I will add them here (and in Blackboard).

Lecture 1 , Lecture 2 , Lecture 3 , Lecture 4 , Lecture 5 , Lecture 6 , Lecture 7 , Lecture 8 , Lecture 9 , Lecture 10 , Lecture 11 , Lecture 12, Part 1 , Lecture 12, Part 2 , Lecture 15 , Lecture 16 , Lecture 17 , Lecture 18 , Lecture 19 , Lecture 20 , Lecture 21 , Lecture 22

Exercise Sheet 1

Solutions 1

Exercise Sheet 2

Solutions 2

Exercise Sheet 3

Solutions 3

Exercise Sheet 4

Solutions 4

Exercise Sheet 5

Solutions 5

Exercise Sheet 6

Solutions 6

Exercise Sheet 7

Solutions 7

Exercise Sheet 8

Solutions 8

Exercise Sheet 9 Only the first question - "Sheet 8, Question 7". The remaining material has not been covered and is not examinable.

Solutions 9 This is for information only - the material on total differentials is not examinable.

Coursework exams Here is the first coursework exam, which was held on Friday 22nd February at the time (9.00) and place of the usual OB2 tutorial. Plus Solutions. Plus some Comments on common errors. The second took place on Friday 26th April. Here are the Solutions and comments.

The standard formula booklet (copied here) will be supplied in the exam.

Here is the 2012 Exam (earlier ones are available at the course page School of Mathematics OB2 Course materials webpage).

Notes on Complex Numbers: I spent the time in the lectures on Complex Numbers on explanation and examples, rather than giving clean notes, so here are some old lecture notes of mine on Complex Numbers, which I've scanned. I think the last couple of pages might be missing - it would have been natural to give more applications of De Moivre's Theorem - but you can find some such examples on pages 32-34 of these notes (from another course on Complex Numbers that I taught).