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A general idea in "for all" statements

A general idea in "for all" statements

I have a proposition for the page Proving "for all" statements.

There is a more general approach/view, which generalize induction and the articles in the subsection Prove the result for some cases and deduce it for the rest.

It is the following: If you have to show a statement  A(x) it is useful to exploit the structure of the set X.

If there are for example some operations T_1,...T_k acting on X such that you can write every x \in X as T_{i_1}^{n_1}...T_{i_m}^{n_m}x_0 with a special point x_0 and you can show:

1) A(x_0) is true

2) If you apply one of the transformations T_i to an y \in X where A(y) is true then A(T_iy) remains true.

Then you have shown your statement. (Induction works like this)

Second example: If your set X is partially ordered, such that every nonempty subset Y \subset X has an element y_0 \in Y which is minimal in the following sence: If y_0 and y \in Y are comparable, then y_0 \leq y. Then you can show the statement by assuming that the set of counterexamples is not empty and look at one of the minimal counterexamples. If one can show, that there is no minimal counterexample, the proof is complete.

I hope it is clear what I want to say... Such arguments are used for example in the theory of finite groups.

There are many more special cases of this principle, but I think, there should be an article which summarize the idea.

I do not feel capable to make an article myself because of my bad english skills, but probably someone else can...