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Exception to a proposed general rule
A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated
Counterexample
Informal logical fallacy
in which one modifies a prior claim in response to a counterexample by asserting the counterexample is excluded by definition. Rather than admitting error
No_true_Scotsman
Smallest example which falsifies a claim
In mathematics, a minimal counterexample is the smallest example which falsifies a claim. It is also sometimes called a minimal criminal, smallest criminal
Minimal_counterexample
Philosophical argument
known as Frankfurt counterexamples or Frankfurt-style cases) were presented by philosopher Harry Frankfurt in 1969 as counterexamples to the principle of
Frankfurt_cases
Book by Lynn Steen
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
Counterexamples_in_Topology
Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by
Witsenhausen's_counterexample
Method of proof in mathematics
disproved by giving a counterexample, as in classical mathematics. However, it is also possible to give a Brouwerian counterexample to show that the statement
Constructive_proof
Technique for symbolic model checking and logic calculi
Counterexample-guided abstraction refinement (CEGAR) is a technique for symbolic model checking. It is also applied in modal logic tableau calculi algorithms
Counterexample-guided abstraction refinement
Counterexample-guided_abstraction_refinement
Counterexample in algebraic geometry
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
Hironaka's_example
Potential counterexample to the generalized Riemann hypothesis
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
Siegel_zero
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a time
Fubini's_theorem
Are certain algebras finitely generated
1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants
Hilbert's_fourteenth_problem
Large number coined by Ronald Graham
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.
Graham's_number
American mathematician
constructed a counterexample that disproved it. Her work involved using fractals and other tools and originally resulted in a more complex counterexample before
Hannah_Cairo
Conjecture in group theory
conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on
Andrews–Curtis_conjecture
Disproven conjecture in number theory
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
Pólya_conjecture
Impossibility for separate objects to have all their properties in common
properties. Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it
Identity_of_indiscernibles
Ability to examine a theory by experimentation
that counterexamples to the hypothesis are logically possible. The practical feasibility of observing a reproducible series of such counterexamples if they
Testability
Japanese mathematician
counterexamples for some seemingly plausible statements in commutative algebra and algebraic geometry, earning him the nickname "Mr. Counterexample"
Masayoshi_Nagata
Theorem in probability theory
(Xn)n∈ N {\displaystyle \mathbb {N} } and the number N of terms; see the counterexample below for the necessity. Note that assumption (2) is satisfied when
Wald's_equation
Unproven conjecture in graph theory
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
Erdős–Gyárfás_conjecture
Philosophical problem about what constitutes knowledge
knowledge. Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held justified true belief
Gettier_problem
Subset of a topological space whose closure is compact
non-compact homogeneous spaces (specifically spaces of lattices). As a counterexample take any finite neighbourhood of the particular point of an infinite
Relatively_compact_subspace
Ordered field where every nonnegative element is a square
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
Euclidean_ordered_field
Transmission of knowledge and skills
occasionally fall outside their parameters. The difficulty of dealing with counterexamples not covered by precise definitions can be avoided by offering less
Education
Type of category in mathematics
In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category
Closed_monoidal_category
Proposition in mathematics that is unproven
insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes
Conjecture
Problem in graph theory
widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples. The problem of finding Hamiltonian paths
Lovász_conjecture
American mathematician (1927–2010)
cover this prize. A counterexample will actually earn its discoverer $620, because Samuel Wagstaff offers $100 for a counterexample or a proof, and Carl
John_Selfridge
Process of making software accessible worldwide
In computing, internationalization and localization (American) or internationalisation and localisation (Commonwealth), often abbreviated i18n and l10n
Internationalization and localization
Internationalization_and_localization
Economic observation that capital does not flow from rich to poor countries
In economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed countries to developing countries despite
Lucas_paradox
2013 book by J. M. Stoyanov
Counterexamples in Probability is a mathematics book by Jordan M. Stoyanov. Intended to serve as a supplemental text for classes on probability theory
Counterexamples in Probability
Counterexamples_in_Probability
On lengths of shortest paths in convex polytopes
43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as
Hirsch_conjecture
On the existence of hyperplanes separating disjoint convex sets
in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness. This version of the separation theorem does generalize
Hyperplane_separation_theorem
1986 book by Romano and Siegel
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
Counterexamples in Probability and Statistics
Counterexamples_in_Probability_and_Statistics
Counterexample to the converse of the intermediate value theorem
mathematical function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it
Conway's_base_13_function
Philosophical problem articulated by David Hume
life), a valid body of knowledge of right desire is generated. Several counterexamples have been offered by philosophers claiming to show that there are cases
Is–ought_problem
Norio Iwase gave a counterexample to the general case in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with X a closed
Ganea_conjecture
Published by Tutte in 1946, it is the first counterexample constructed for this conjecture. Other counterexamples were found later, in many cases based on
Tutte_graph
a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other
Whitehead_conjecture
Rule of logical inference
paradoxes of material implication). The general form of McGee-type counterexamples to modus ponens is simply P , P → ( Q → R ) {\displaystyle P,P\rightarrow
Modus_ponens
Mathematical problem solved in 1967
functions that will converge to the counterexample to the common fixed point problem. Although the discovery of counterexamples by Boyce and Huneke meant that
Common_fixed_point_problem
Pathological embedding of the sphere in 3D space
realizing his error, he constructed the horned sphere as a definitive counterexample. Alexander's genius was in realizing that the "wildness" Antoine had
Alexander_horned_sphere
1976 book by Imre Lakatos
counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample'
Proofs_and_Refutations
Paradox arising from an incorrect proof
Two differently colored horses, providing a counterexample to the general theorem
All_horses_are_the_same_color
American philosopher (1929–2023)
action. In the field of ethics, Frankfurt gave various influential counterexamples, so-called Frankfurt cases, against the principle that moral responsibility
Harry_Frankfurt
Continuous function that is not absolutely continuous
that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity
Cantor_function
statistically independent. However, this is untrue, as can be demonstrated by counterexample. Likewise, it is sometimes mistakenly thought that a linear combination
Misconceptions about the normal distribution
Misconceptions_about_the_normal_distribution
Expansion of auto-oriented, low-density development in suburbs
Urban sprawl (also known as suburban sprawl or urban encroachment) is defined as "the rapid expansion of the geographic extent of cities and towns, often
Urban_sprawl
Type of infinite group in group theory
is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem
Tarski_monster_group
constant mean curvature, discovered by Henry C. Wente (1986). It is a counterexample to the conjecture of Heinz Hopf that every closed, compact, constant-mean-curvature
Wente_torus
Ring in abstract algebra
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided)
Artinian_ring
School of thought within decision theory
probability less than 1, leading to regress. There are innumerable "counterexamples" where, it is argued, a straightforward application of CDT fails to
Causal_decision_theory
y)=(2x^{2}-y)(y-x^{2}).} It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of
Peano_surface
Discussions and claims of differences in intelligence along racial lines
Du Bois, and the poet Paul Laurence Dunbar stood as high-profile counterexamples to widespread stereotypes of black intellectual inferiority. In Britain
Race_and_intelligence
Treating art as a natural phenomenon
definitions are composed of necessary and sufficient conditions, and a single counterexample overthrows such a definition. Theorizing about art, on the other hand
Theory_of_art
Written work often reflecting the author's personal point of view
Understanding and Thomas Malthus's An Essay on the Principle of Population are counterexamples. In some countries, such as the United States and Canada, essays have
Essay
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite
Ellingham–Horton_graph
Graph theory concept
the conjecture is false, K1,2,2,2 would necessarily be its smallest counterexample. A related conjecture by Michael Fellows, now solved, concerns planar
Planar_cover
Counterintuitive mathematical object
such as the Black-Scholes model in finance. Counterexamples in Analysis is a whole book of such counterexamples. Another example of pathological function
Pathological_(mathematics)
Delivering multicast packets based on source and destination addressing
Source-specific multicast (SSM) is a method of delivering multicast packets in which the only packets that are delivered to a receiver are those originating
Source-specific_multicast
On unit fractions adding to 4/n
n} that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified
Erdős–Straus_conjecture
Probabilistic primality testing algorithm
Pomerance, Selfridge, and Wagstaff offered $30 for the discovery of a counterexample, that is, a composite number that passed this test, or the production
Baillie–PSW_primality_test
Distinction between what is and what ought to be
is how Hume and the later positivists conceived of facts. Several counterexamples have been offered by philosophers claiming to show that there are cases
Fact–value_distinction
Disproven graph theory
Tutte (1946), who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38
Tait's_conjecture
Method of media consumption
Database consumption (Japanese: データベース消費, romanized: dētabēsu shōhi) refers to a way of content consumption in which people do not consume a narrative
Database_consumption
Singularities of holomorphic functions extend infinitely outward
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of
Hartogs's_extension_theorem
Problem in number theory on equal totients
exists a counterexample to the conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples. Although
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Fundamental theorem in mathematical logic
in a model, then one of the model's natural numbers is a counterexample. If this counterexample existed within the standard natural numbers, its existence
Gödel's_completeness_theorem
Cubic graph with 10 vertices and 15 edges
and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after
Petersen_graph
Chinese mathematician
are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's Last Theorem. Zhi-Hong Sun's homepage Archived 2006-07-25
Zhi-Hong_Sun
The area cut off by a secant of a smooth convex oval is not an algebraic function
"oval" means merely a continuous closed convex curve, then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate y2 = x2 − x4
Newton's_theorem_about_ovals
Proposed constraint in theoretical linguistics
languages form a natural class of counterexamples to the FOFC. Thus, it must be then investigated whether such counterexamples do indeed violate FOFC, and if
Final-over-final_constraint
Three groups
Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied
Thompson_groups
Conjecture in number theory
a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently
Beal_conjecture
Mathematical concept
statement that is not universally true for all ordinals must have a minimal counterexample. In fact, this principle is also true for arbitrary well-ordered sets
Transfinite_induction
In algebraic geometry, a divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a
Divisorial_scheme
Class of organic compounds
the benzene rings. Most aromatic hydrocarbons are benzenoid. Notable counterexamples are cyclooctadecanonaene, azulene and trans-bicalicene. Quinoid Aromatic
Benzenoid
Theorem in Riemannian geometry
the closed interval [ 1 , 4 ] {\displaystyle [1,4]} . The standard counterexample is complex projective space with the Fubini–Study metric; sectional
Sphere_theorem
Category of mathematical proof
possible counterexamples to be invalid: at least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility
Proof_of_impossibility
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
Naimark's_problem
Topology on the real numbers
set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology
Lower_limit_topology
On dissections between polyhedra
always possible. His student Max Dehn confirmed the conjecture with a counterexample. The formula for the volume of a pyramid, one-third of the product of
Hilbert's_third_problem
Local-global result for when an element in a number field is an nth power
appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself
Grunwald–Wang_theorem
Index of articles associated with the same name
include the following: Brazilian Journal of Probability and Statistics Counterexamples in Probability and Statistics Probability and Mathematical Statistics
Probability_and_statistics
1994 suicide of Nirvana singer and guitarist
high dose could be survived, although he does not rule out whether a counterexample might exist (for updated information on the question of how to interpret
Suicide_of_Kurt_Cobain
counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a C 2 + δ {\displaystyle C^{2+\delta }} counterexample for
Seifert_conjecture
Field of knowledge
if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often
Mathematics
Number which when multiplied by x equals 1
associativity. In the absence of associativity, the sedenions provide a counterexample. The converse does not hold: an element which is not a zero divisor
Multiplicative_inverse
Theorem in homological algebra
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian
Snake_lemma
Chinese mathematician
appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself
Wang_Xianghao
2023–24 U.S. legal and political dispute
decisions. Graber states that Chase's opinion in Griffin's Case is the only counterexample following the ratification of the 14th Amendment of a court or legislative
Presidential eligibility of Donald Trump
Presidential_eligibility_of_Donald_Trump
Disproven mathematical theory concerning Banach-Tarski and amenable groups
within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide-open problem, the
Von_Neumann_conjecture
Computer science field
in fact, be sufficient to prove e.g. properties of mutual exclusion. Counterexample-guided abstraction refinement (CEGAR) begins checking with a coarse
Model_checking
Type of commutative ring in mathematics
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality
Cohen–Macaulay_ring
Topological continuum undefinable as the union of any two proper subcontinua
Indecomposable continua have been used by topologists as a source of counterexamples. They also occur in dynamical systems. A continuum C {\displaystyle
Indecomposable_continuum
Classic counterexample in topology
Lynn Arthur; Seebach, J. Arthur Jr (1995) [First published 1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Mineola, NY: Dover Publications
Infinite_broom
Efficiency leads to increased demand
increased demand for food. (Demand for food is inelastic.[source needed. Counterexample—food consumed has increased in resource intensity, e.g. increased meat
Jevons_paradox
Unsolved problem in graph theory
constructed a counterexample with 46 vertices; other researchers later found even smaller counterexamples. However, none of these known counterexamples is bipartite
Barnette's_conjecture
COUNTEREXAMPLE
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Boy/Male
African, Arabic, Hindu, Indian, Iranian, Marathi, Muslim, Parsi, Pashtun, Sindhi, Swahili
Priestly; Happy; Fortunate; Prosperous; Lucky; Rivulet; Blissful; Auspicious
Boy/Male
Hindu, Indian
Lord Krishna
Girl/Female
Hindu, Indian
Godess Laxmi
Girl/Female
Hindu
A fairy-tale princess
Boy/Male
Hindu
Another name of Lord Murugan
Girl/Female
Muslim/Islamic
Proper name
Male
Spanish
Spanish form of Hebrew Efrayim, EFRAÃN means "double-land; twin-land."
Girl/Female
Hindu
Goddess Laxmi, Consort of Lord Krishna
Girl/Female
Indian
Jasmine, Flower
Girl/Female
Indian, Kannada, Tamil
Name of a Flower; Jasmine
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