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11

Properties of finite sets

There are a number of special techniques available for proving results which involve finite sets. It is clearly possible to determine whether or not a finite set contains an element with a certain property simply by trying each element in turn. More interestingly it is possible on occasion to use ‘counting arguments’ to demonstrate the existence of an element with a certain property without considering the properties of individual elements at all. We consider these techniques in this chapter.

### 11.1 The pigeonhole principle

We begin by recalling the key step used in the last chapter to prove that the cardinality of a finite set is a well-defined concept. We left for this chapter the proof of the following result. ...

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