Please forward this error screen to 199. Since 10 is pigeonhole principle in discrete mathematics pdf than 9, the pigeonhole principle says that at least one hole has more than one pigeon.
This theorem is exemplified in real life by truisms like “in any group of three gloves there must be at least two left gloves or two right gloves”. It is an example of a counting argument. London who have the same number of hairs on their heads.
For this reason it is also commonly called Dirichlet’s box principle or Dirichlet’s drawer principle. This should not be confused with Dirichlet’s principle, a term introduced by Riemann that refers to the minimum principle for harmonic functions. The principle has several generalizations and can be stated in various ways. To do so requires the formal statement of the pigeonhole principle, which is “there does not exist an injective function whose codomain is smaller than its domain”.
Advanced mathematical proofs like Siegel’s lemma build upon this more general concept. Dirichlet published his works in both French and German.
The strict original meaning of either the German Schubfach, or the French tiroir, corresponds to the English drawer, an open-topped box that can be slid in and out of the cabinet that contains it. These terms were morphed to the word pigeonhole, standing for a small open space in a desk, cabinet, or wall for keeping letters or papers, metaphorically rooted in the structures that house pigeons. Considering the fact that Dirichlet’s father was a postmaster, necessarily best acquainted to furniture of type pigeonhole, common for sorting letters in his business, the translation by pigeonholes may be a perfect transfer of Dirichlet’s terms of understanding. The meaning, referring to some furniture features, has since been strongly overtaken and is fading, especially among those who do not speak English natively, but as a lingua franca in the scientific world, in favour of the more pictorial interpretation, literally involving pigeons and holes.
It is interesting to note that the suggestive, though not misleading interpretation of “pigeonhole” as “dovecote” has lately found its way back to a German “re-translation” of the “pigeonhole”-principle as the “Taubenschlag”-principle. Assume a drawer contains a mixture of black socks and blue socks, each of which can be worn on either foot, and that you are pulling a number of socks from the drawer without looking.
What is the minimum number of pulled socks required to guarantee a pair of the same color? Either you have three of one color, or you have two of one color and one of the other. In this application of the principle, the ‘hole’ to which a person is assigned is the number of hands shaken by that person. We can demonstrate there must be at least two people in London with the same number of hairs on their heads as follows.