About: Minimal counterexample

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In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite propert

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rdfs:label	Minimal counterexample (en)
rdfs:comment	In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite propert (en)
dcterms:subject	Mathematical proofs Mathematical terminology
Wikipage page ID	965376 (xsd:integer)
Wikipage revision ID	1098094380 (xsd:integer)
Link from a Wikipage to another Wikipage	Proof by infinite descent Mathematical induction Mathematics Fundamental theorem of arithmetic Cyclic group Feit–Thompson theorem Proof by contradiction Mathematical proofs Counterexample Mathematical terminology Classification of finite simple groups Natural number Solvable group Proof By Induction Well-ordered induction Well-ordering
sameAs	Minimal counterexample Minimal counterexample Minimal counterexample Minimal counterexample
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date	December 2021 (en)
reason	Is the tense correct here? (en)
has abstract	In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true. If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof. The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Minimal_counterexample
is Link from a Wikipage to another Wikipage of	Proof by infinite descent Feit–Thompson theorem Counterexample Minimal criminal
is Wikipage redirect of	Minimal criminal
is foaf:primaryTopic of	wikipedia-en:Minimal_counterexample

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