Search results
Results From The WOW.Com Content Network
Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, [2] it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip ("drawer principle" or "shelf principle").
The existence of these polynomials was proven by Axel Thue; [1] Thue's proof used what would be translated from German as Dirichlet's Drawers principle, which is widely known as the Pigeonhole principle. Carl Ludwig Siegel published his lemma in 1929. [2] It is a pure existence theorem for a system of linear equations.
As an example, exponential proof size lower-bounds in resolution for the pigeon hole principle imply that any algorithm based on resolution cannot decide TAUT or SAT efficiently and will fail on pigeon hole principle tautologies. This is significant because the class of algorithms based on resolution includes most of current propositional proof ...
The pigeonhole principle states that if a items are each put into one of b boxes, where a > b, then one of the boxes contains more than one item. Using this one can, for example, demonstrate the existence of some element in a set with some specific properties.
Consider the following theorem (which is a case of the pigeonhole principle): If three objects are each painted either red or blue, then there must be at least two objects of the same color. A proof: Assume, without loss of generality, that the first object is red.
The proof that the language of balanced (i.e., properly nested) parentheses is not regular follows the same idea. Given p {\displaystyle p} , there is a string of balanced parentheses that begins with more than p {\displaystyle p} left parentheses, so that y {\displaystyle y} will consist entirely of left parentheses.
This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. [2] The method extends to simultaneous ...
Using the Pigeonhole Principle Thue, and later Siegel, managed to prove the existence of auxiliary functions which, for example, took the value zero at many different points, or took high order zeros at a smaller collection of points.