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:''This article is about the type of mathematical proof. For the method of calculating limits, see Method of 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.
To prove that every cube number 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.
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 because these proofs feel inelegant—they leave an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. 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.
★ Case analysis
★ Computer-assisted proof
'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.
| Contents |
| Example |
| How many cases? |
| See also |
Example
To prove that every cube number 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 because these proofs feel inelegant—they leave an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. 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
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