Pigeonhole principle

Statement

Let $A$ and $B$ be finite sets with $|A| > |B|$ , and let $f : A \to B$ be a function. Then there are elements $a \neq a'$ of $A$ with $f(a) = f(a')$ . In other words, $f$ is not injective.

Remarks

In the usual image, if $n + 1$ pigeons sit in $n$ holes, then some hole has at least two pigeons. Example: among any $13$ people, two have their birthdays in the same month. The generalized pigeonhole principle gives a fibre with more than $n$ elements. The infinite pigeonhole principle is a version for infinite sets.

Reading path

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Cardinality of a finite set definition
Pigeonhole principle statement

Where this leads

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Tag	`0HGVCWQGY3`
URI	`m2://0HGVCWQGY3`
Kind	statement, theorem
Writing	W1 stated
MSC	05A05
Status	This copy of the site has no build data. Run `m2 status 0HGVCWQGY3`.
Uses	Cardinality of a finite set
Same as	Wikidata Q188276 Wikipedia Pigeonhole principle

Pigeonhole principle

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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