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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical proof | 3/4 | https://en.wikipedia.org/wiki/Mathematical_proof | reference | science, encyclopedia | 2026-05-05T07:24:53.330614+00:00 | kb-cron |
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color 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.
=== Closed chain inference ===
A closed chain inference shows that a collection of statements are pairwise equivalent. In order to prove that the statements
φ
1
,
…
,
φ
n
{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}
are each pairwise equivalent, proofs are given for the implications
φ
1
⇒
φ
2
{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}
,
φ
2
⇒
φ
3
{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}
,
…
{\displaystyle \dots }
,
φ
n
−
1
⇒
φ
n
{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}
and
φ
n
⇒
φ
1
{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}
. The pairwise equivalence of the statements then results from the transitivity of the material conditional.
=== Probabilistic proof ===
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems. In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one. A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward the Collatz conjecture shows how far plausibility is from genuine proof, as does the disproof of the Mertens conjecture. While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality) are as good as genuine mathematical proofs.
=== Combinatorial proof ===
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
=== Nonconstructive proof ===
A nonconstructive proof establishes that a mathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that
a
b
{\displaystyle a^{b}}
is a rational number. This proof uses that
2
{\displaystyle {\sqrt {2}}}
is irrational (an easy proof is known since Euclid), but not that
2
2
{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
is irrational (this is true, but the proof is not elementary).
Either
2
2
{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
is a rational number and we are done (take
a
=
b
=
2
{\displaystyle a=b={\sqrt {2}}}
), or
2
2
{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
is irrational so we can write
a
=
2
2
{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
and
b
=
2
{\displaystyle b={\sqrt {2}}}
. This then gives
(
2
2
)
2
=
2
2
=
2
{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}
, which is thus a rational number of the form
a
b
.
{\displaystyle a^{b}.}
=== Statistical proofs in pure mathematics ===
The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic number theory or analytic number theory. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also the "Statistical proof using data" section below.
=== Computer-assisted proofs ===
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, automated theorem provers and proof assistants are now used to prove theorems and carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.