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How to Think Like a Mathematician by Kevin Houston

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CHAPTER 22

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Techniques of proof II: Proof by cases

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Little by little does the trick.

Aesop, The Crow and the Water Jar

We have already seen that x = y can be proved by showing that xy and xy. In other words we have broken the problem into two cases. This is a very common procedure in mathematics. Often we can break down some problem and tackle each case individually utilizing different methods, or even the same methods, in each case.

The famous Four Colour Problem is that given a map (of the geographical kind) we need only four colours to colour ...

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