Pigeonhole Principle
Theorem
Let $S$ be a finite set whose cardinality is $n$.
Let $S_1, S_2, \ldots, S_k$ be a partition of $S$ into $k$ subsets.
Then:
where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Corollary
If a set of $n$ distinct objects is partitioned into $k$ subsets, where $0 < k < n$, then at least one subset must contain at least two elements.
Proof
Aiming for a contradiction, suppose no subset $S_i$ of $S$ has as many as $\ceiling {\dfrac n k}$ elements.
Then the maximum number of elements of any $S_i$ would be $\ceiling {\dfrac n k} - 1$.
So the total number of elements of $S$ would be no more than $k \paren {\ceiling {\dfrac n k} - 1} = k \ceiling {\dfrac n k} - k$.
There are two cases:
Suppose $k \divides n$.
Then $\ceiling {\dfrac n k} = \dfrac n k$ is an integer and:
- $k \ceiling {\dfrac n k} - k = n - k$
Thus:
- $\displaystyle \sum_{i \mathop = 1}^k \card {S_i} \le n - k < n$
This contradicts our assumption that no subset $S_i$ of $S$ has as many as $\ceiling {\dfrac n k}$ elements.
Next, suppose that $k \nmid n$.
Then:
- $k \ceiling {\dfrac n k} - k < \dfrac {k \paren {n + k} } k - k = n$
and again this contradicts our assumption that no subset $S_i$ of $S$ has as many as $\ceiling {\dfrac n k}$ elements.
Either way, there has to be at least $\ceiling {\dfrac n k}$ elements in at least one $S_i \subseteq S$.
$\blacksquare$
Source of Name
It is known as the Pigeonhole Principle because of the following.
Suppose you have $n + 1$ pigeons, but have only $n$ holes for them to stay in.
By the Pigeonhole Principle at least one of the holes houses $2$ pigeons.
Historical Note
The Pigeonhole Principle has been attributed to Johann Peter Gustav Lejeune Dirichlet.
It is also known as Dirichlet's Box (or Drawer) Principle, or, as Dirichlet named it, Schubfachprinzip (drawer principle or shelf principle).
In Russian and some other languages, it is known as the Dirichlet principle or Dirichlet's principle, which name ambiguously also refers to the minimum principle for harmonic functions.
Also see
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.18$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 2.2$: Isomorphic Graphs: Problem $23$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Dirichlet's principle
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Pigeonhole Principle
- Weisstein, Eric W. "Dirichlet's Box Principle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DirichletsBoxPrinciple.html