A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of...

mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which...

in which one modifies a prior claim in response to a counterexample by asserting the counterexample is excluded by definition. Rather than admitting error...

Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working...

conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős. If the conjecture is false, a counterexample would take the form of a...

known as Frankfurt counterexamples or Frankfurt-style cases) were presented by philosopher Harry Frankfurt in 1969 as counterexamples to the principle of...

1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants...

Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by...

disproved by giving a counterexample, as in classical mathematics. However, it is also possible to give a Brouwerian counterexample to show that the statement...

the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli...

scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped. For quasi-projective...

non-compact homogeneous spaces (specifically spaces of lattices). As a counterexample take any finite neighbourhood of the particular point of an infinite...

contain no such subgraph if, for example, the bottom edge in the present subgraph were replaced by a blue edge – thus proving by counterexample that N* > 3....

generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by ℵ 1 {\displaystyle...

counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a C 2 + δ {\displaystyle C^{2+\delta }} counterexample for...

Counterexample-guided abstraction refinement (CEGAR) is a technique for symbolic model checking. It is also applied in modal logic tableau calculi algorithms...

after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions...

knowledge. Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held justified true belief...

Counterexamples in Probability and Statistics is a mathematics book by Joseph P. Romano and Andrew F. Siegel. It began as Romano's senior thesis at Princeton...

within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide-open problem, the...

more accurately called "Pólya's problem". The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true...

conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on...

insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes...

In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some...

set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology...

In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of...

Tutte (1946), who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38...

In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category...

that counterexamples to the hypothesis are logically possible. The practical feasibility of observing a reproducible series of such counterexamples if they...

in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness. This version of the separation theorem does generalize...