## 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
Induction
The End

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Apart from understanding the statements of abstract theorems (see Formulating mathematics), you should become used to proving them rigorously. The most important theorems will be proved in books and lectures, but you may be asked to reproduce the proofs in examinations. Moreover, you will frequently be set problems which require you to prove abstract statements which you have not seen before. In all cases, the aim will be to write out accurate and efficient proofs. There are three basic skills concerning proofs:

(i)
identifying the essential points of a standard proof,
(ii)
constructing proofs of facts which were previously unknown to you,
(iii)
setting out proofs accurately, clearly, and efficiently.
We have discussed (i) in Part 1: Studying the theory, and (iii) in Part 1: Writing mathematics. Many of the principles described in Part 1: Problem solving are applicable to (ii) (see Experimentation), but this chapter will be devoted to a more detailed discussion of (ii). The examples in Making the proof precise and Examples will also serve to illustrate (iii).
Design: Paul Gartside,
Content: Prof. C. Batty,
December 1999.
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