Study Guide: Part 1.2.4

Summary

1.: When writing out a solution, ensure that your argument is complete and accurate, and that, when the symbols are converted into English, what you have written makes full sense grammatically and is logically correct.
2.: When writing out a solution, pay attention to the points listed below ((i)-(vii)).
3.: Good punctuation, particularly the use of commas and brackets, may remove ambiguities.

The third aspect of doing mathematics is the written exposition of your work. As a student, you should always aim to write out your work in a complete, accurate, and clear fashion. This applies equally to writing out theory and to solutions of problems.

There are several reasons why good exposition is important. Firstly, when you come to write down details of problems, you may find some difficulties which you had overlooked in your rough working. Secondly, any tutor or examiner who is going to read your work will need evidence that you really understood what you were doing. Thirdly, you should aim to acquire the ability to communicate mathematics to other people; this will obviously be a valuable skill if you follow a career which involves mathematics, and it will also enhance your ability to write unambiguously and logically in any occupation. In mathematics, accuracy is much more important than literary quality!

Although your work may be read only by your tutor, it is a good idea, when you are writing your solutions, to imagine that you are writing for a reader who has a similar level of mathematical knowledge to yourself but who has not encountered the argument which you are writing down. Such a reader should be able to understand what you have written without danger of ambiguity, to see that it is correct, and to see why it is correct.

The English language (in common with other languages) is prone to ambiguity, and mathematicians have to be particularly careful to use language (and symbols, which are part of mathematical language) in an unambiguous way. Good punctuation can be very helpful; for example, commas can be used to separate clauses. Sometimes, mathematicians go beyond normal punctuation by using brackets to remove ambiguity. The use of brackets inside algebraic expressions is commonplace (consider the difference in meaning between (xy)+z and x(y+z); it is a mathematical convention that the unbracketed expression xy+z means (xy)+z, so if we mean x(y+z), we have to include the brackets). However, brackets can also be used around entire clauses, to remove ambiguity. For example, the statement:

x = 0 and y = 1 or z = 2 and t = 3,

is highly ambiguous, but the following are not:

[x = 0 and y = 1] or [z = 2 and t = 3],

x = 0, and [y = 1 or z = 2], and t = 3.

When you write out your final solution, make sure that it is fully comprehensible, and that each step follows logically from previous ones. What you write should be clear, intelligible and unambiguous, as well as being logically correct and mathematically accurate. In particular, when all the symbols are converted into English, your script should consist of complete sentences. You should pay particular attention to points such as the following:

(i): If you are arguing by contradiction or induction (see Part2: Proofs by contradiction and Part 2: Proofs by induction, say so clearly.
(ii): If you make some assumptions, say so clearly, by including phrases such as ``Suppose that ...''.
(iii): Be careful to use phrases such as ``if'', ``only if'', ``if and only if'', correctly (see Implications). If you use symbols such as , , and Û, make sure that they go in the direction which is both logically correct and relevant to the problem (this may be the opposite of the direction in your rough working).
(iv): Include phrases such as ``and'', ``or'', ``for all'', ``for some'' (see Part 2: And & Or and Part 2: For all & There exists;) ; you cannot expect the reader to know which you mean if you omit these phrases.
(v): Make sure that your argument shows correctly which quantities depend on which other quantities (see Part 2: Dependence).
(vi): If the statement which you are writing down depends on something several lines earlier, make this clear to the reader, for example by labelling the earlier statement with (*) and writing ``It follows from (*) that ...''.
(vii): If your proof involves a division into two or more cases, make it clear what the different cases are (see Part 2: And & Or).

See Part 2: The formulation of mathematical statements for further discussion of several of these points.

Part 1

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