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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
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.
For example, a particular statement may be shown to imply the law of the excluded middle. An example of a Brouwerian counterexample of this type is Diaconescu's theorem, which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiom of choice implies the law of excluded middle in such systems.
A proof by example is an argument whereby a statement is not proved but instead illustrated by an example. If done well, the specific example would easily generalize to a general proof. by inspection A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction.
A counterexample is provided by a cube where one face is replaced by a square pyramid: this elongated square pyramid is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive.
Petersen graph as Kneser graph ,. The Petersen graph is the complement of the line graph of .It is also the Kneser graph,; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other.
The latter has been positively solved for an extremely large class of groups, including for example all hyperbolic groups. The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two (as witnessed by the earlier-mentioned counter-example).
Pappus theorem: proof. If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique.