## 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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In pure mathematics, a typical problem is quite theoretical, requiring you to prove something of greater or lesser generality. We saw in Part 1: Problem solving that problem-solving involves three stages which in this context can best be described as:

(i)
understanding the problem (understanding the statement, and what is to be proved);
(ii)
experimentation (to find a method which looks likely to work);
(iii)
making the proof precise (making sure that the proof does work, and writing it out accurately and completely).

We shall discuss these three stages in turn by way of an example.

First, read through the example. If there is anything you dont understand, dont worry, all will be explained.

Example 0. Let A, B, and C be sets. Prove that (A ÈB) Ç(AÈC) = A È(B ÇC).

Design: Paul Gartside,
Content: Prof. C. Batty,
December 1999.
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