Proof by exhaustion

Proof by exhaustion

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. A proof by exhaustion contains two stages:

*A proof that the cases are exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases.
*A proof of each of the cases.

In contrast, the method of exhaustion of Eudoxus of Cnidus was a geometrical and essentially rigorous way of calculating mathematical limits.

Example

To prove that every integer that is a perfect cube is either a multiple of 9, or 1 more, or 1 less than a multiple of 9.

Proof:
Each cube number is the cube of some integer "n". This integer "n" is either a multiple of 3, or 1 more or 1 less than a multiple of 3. So these 3 cases are exhaustive:
*Case 1: If "n" = 3"p", then "n"³ = 27"p"³, which is a multiple of 9.
*Case 2: If "n" = 3"p"+1, then "n"³ = 27"p"³+27"p"²+9"p"+1, which is 1 more than a multiple of 9. For instance, if "n" = 4 then "n"³ = 64 = 9x7+1.
*Case 3: If "n" = 3"p"−1, then "n"³ = 27"p"³−27"p"²+9"p"−1, which is 1 less than a multiple of 9. For instance, if "n" = 5 then "n"³ = 125 = 9x14−1.

How many cases?

There is no upper limit to the number of cases allowed in a proof by exhaustion. Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving an endgame puzzle in chess might involve considering a very large number of possible positions in the game tree of that problem.

The first proof of the four colour theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

Mathematicians prefer to avoid proofs with large numbers of cases.Such proofs feel inelegant to them. A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection.Other types of proofs -- such as proof by induction (mathematical induction) -- are considered more elegant.However, there are some important theorems for which no other method of proof has been found.

As well as the four colour theorem, other examples of large proofs by exhaustion are:

* The proof that there is no finite projective plane of order 10.
* The classification of finite simple groups.
* The Kepler conjecture.

See also

* Case analysis
* Computer-assisted proof


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • proof by exhaustion — noun The indirect verification or falsification of a statement by the verification or falsification of each of the finite number of cases which arise therefrom …   Wiktionary

  • Proof by verbosity — is a term used to describe an excessively verbose mathematical proof that may or may not actually prove the result. Such proofs are most often presented by students who don t fully grasp the concepts they are writing about. Students presenting… …   Wikipedia

  • Mathematical proof — In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.[1][2] Proofs are obtained from deductive reasoning, rather than from inductive or empirical… …   Wikipedia

  • Method of exhaustion — This article is about the method of finding the area of a shape using limits. For the method of proof, see Proof by exhaustion. The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape… …   Wikipedia

  • Computer-assisted proof — A computer assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer aided proofs to date have been implementations of large proofs by exhaustion of a mathematical theorem. The idea is to use a… …   Wikipedia

  • Direct proof — In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. In order …   Wikipedia

  • Kepler conjecture — In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing …   Wikipedia

  • Exhaust — may refer to:In mathematics: *Proof by exhaustion, proof by examining all individual cases *Exhaustion by compact sets, in analysis, a sequence of compact sets that converges on a given set *Collectively exhaustive, in probability and set theory …   Wikipedia

  • Experimental mathematics — For the mathematical journal of the same name, see Experimental Mathematics (journal) Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and… …   Wikipedia

  • Полный перебор — У этого термина существуют и другие значения, см. Перебор. Полный перебор (или метод «грубой силы», англ. brute force)  метод решения математических задач. Относится к классу методов поиска решения исчерпыванием всевозможных… …   Википедия

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”