## Part 2

### What is studied

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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Summary
1.
Look up the definitions of any terms and symbols about which you are doubtful.

2.
Identify the hypotheses, write them down, and write down any reformulations which appear useful.

3.
Identify the conclusions to be proved, write them down, separating them into various parts; write down any reformulations which appear promising.

The main things to be done here are:

(i)
Look up the definition of any terms (words or symbols) about which you are uncertain. Likely places to look are your lecture notes, and textbooks; the index of a textbook usually includes a reference to each definition; some textbooks have a separate index of symbols. By the time that you have done this, you should be able to translate the statement into ordinary English.
(ii)
Decide which parts of the statement are hypotheses which you are supposed to assume. Write them down (``We assume that ...''); include any reformulations of the hypotheses which look as though they might be useful; for example, you may replace a technical mathematical term by its definition.
(iii)
Identify the conclusions which you are supposed to prove; write them down (``We have to prove that ...''). If there is more than one part to the conclusion, write them down separately. If appropriate, write down any reformulations of the conclusion which appear promising; for example, it may be useful to put the conclusion in a more mathematical form, by replacing technical terms by their definitions.

In the context of

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

We see:

Definitions: It is likely that the reader is familiar with the notions of union and intersection, and the symbols È, Ç which denote them.

Hypotheses: The only hypothesis is that A, B and C are sets. (It is implicit that they are all considered to be subsets of some universal set.)

Conclusions: We have to prove that (A ÈB) Ç(A ÈC) = A È(B ÇC). To do this, we should show:

(a)
If x Î (A ÈB) Ç(A ÈC), then x Î A È(B ÇC);
(b)
If x Î A È(B ÇC), then x Î (A ÈB) Ç(A ÈC).
Design: Paul Gartside,
Content: Prof. C. Batty,
December 1999.
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