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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This part is the real core to the solution, where you do most of the hard thinking. What you do at this stage may not form part of your final solution, at least not in the form in which you do it at this stage, but it should lead you to see why the statement to be proved is true, and also show you how to put together a rigorous argument. Another description for ``experimentation'' might be ``rough working'', or ``exploration'', but we have in mind not just calculations and algebraic manipulations, but also the whole process of finding a method which works to solve the problem.

The general principles which should apply while exploring the problem have already been listed in Part 1: Problem solving. We repeat some of them here:

(i)

While searching for a method, do not worry about details.

(ii)

Try working backwards from the conclusion.

(iii)

Draw a diagram, if appropriate.

(iv)

Try some special cases, examples, or simpler versions of the problem.

(v)

If appropriate, break the problem up into several cases.

(vi)

Consider the possibility of a proof by contradiction (see Section 5.7).

It is quite likely that your first attempt will not be successful. Keep trying different methods until you find one which works or until you run out of ideas. Look in your lecture notes and books for relevant worked examples. If none of this works, put the problem aside and return to it later.

We now consider the process of experimentation for our Example 0, making use of the clarifications described in Understanding. Of course, what we do here is unlikely to be typical of your work in that we shall not describe many methods which lead to dead ends!

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

Idea:. Let's draw a Venn diagram of (A ÈB) Ç(A ÈC):

 

This is exactly A È(B ÇC), as required. So, a formal argument should work without serious difficulty.