Part 2

New Skills

Formulating Maths
Introduction
Hypotheses &c
Implications
And & or
For all & there exists
Dependence

Proof
Introduction
Counter examples
Constructing proofs
Understanding
Experimentation
Precision
Examples
Assumptions
Contradiction
Induction
The End

 

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Students often worry a great deal about this question, particularly in the early stages of their undergraduate careers. There is indeed some difficulty in judging what it is permissible to assume, but students are inclined to exaggerate the problem. There are two reasons for saying this. One is that tutors will be much less concerned about a solution in which everything is correct except that you have (knowingly) assumed something which you were really intended to prove than about a solution which contains fallacies or non-sequiturs.

The second reason is that, in some of the courses in your first term or two, you will, for a while, be expected to prove things which you have either known for years (such as the existence of nth roots of positive numbers) or which are in some sense obvious (such as the fact that a continuous function which takes positive and negative values must also take the value zero somewhere). Simultaneously, you will be doing other (more applied) courses where these and other familiar or obvious facts are freely used without comment. This can be confusing, but the situation will not last very long. After a while, all courses will be proving facts which you did not know and which are not obvious. When you reach that stage, it should be clearer what you can assume and what you should prove.

Nevertheless, it is possible to give some broad guidelines about this question, all of which are valid in some situations:

(i)

What you are allowed to assume depends on the context; for example, if you are answering questions on a course which is concerned with deriving fundamental properties of the real numbers using only basic axioms, then you should not assume these properties without proof; in courses on mechanics etc., you may assume such properties.

(ii)

When answering tutorial problems, you may assume any results which have been proved in lectures.

(iii)

If you are in doubt, a reasonable compromise is to give a clear statement, without proof, of what you are assuming.

(iv)

If you think carefully about the relationship between various results in lectures and the question you are required to answer, you may find that the point of the question becomes clear and that your doubts about what to assume are resolved.

(v)

Use your common sense.