en.unionpedia.org

Counterexample, the Glossary

A counterexample is any exception to a generalization.^[1]

50 relations: Area, Callicles, Composite number, Conjecture, Contradiction, Control theory, Converse (logic), Counterexamples in Probability, Counterexamples in Probability and Statistics, Counterexamples in Topology, Deductive reasoning, Euclidean plane isometry, Euler's sum of powers conjecture, Exception that proves the rule, Ganea conjecture, Generalization, Geometry, Gorgias (dialogue), Hilbert's fourteenth problem, Hypothesis, Imre Lakatos, J. Arthur Seebach Jr., James Franklin (philosopher), Logic, Loss function, Lynn Steen, Mathematical proof, Mathematics, Minimal counterexample, Natural number, Parity (mathematics), Pólya conjecture, Philosophy, Plato, Prima facie, Prime number, Proofs and Refutations, Rectangle, Rhombus, Rigour, Seifert conjecture, Shape, Shear mapping, Socrates, Square, Squeeze mapping, State variable, Tait's conjecture, Universal quantification, Witsenhausen's counterexample.
Interpretation (philosophy)
Methods of proof

Area

Area is the measure of a region's size on a surface.

See Counterexample and Area

Callicles

Callicles (Καλλικλῆς; c. 484 – late 5th century BC) is thought to have been an ancient Athenian political philosopher.

See Counterexample and Callicles

Composite number

A composite number is a positive integer that can be formed by multiplying two smaller positive integers.

See Counterexample and Composite number

Conjecture

In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Counterexample and conjecture are mathematical terminology.

See Counterexample and Conjecture

Contradiction

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact.

See Counterexample and Contradiction

Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines.

See Counterexample and Control theory

Converse (logic)

In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements.

See Counterexample and Converse (logic)

Counterexamples in Probability

Counterexamples in Probability is a mathematics book by Jordan M. Stoyanov.

See Counterexample and Counterexamples in Probability

Counterexamples in Probability and Statistics

Counterexamples in Probability and Statistics is a mathematics book by Joseph P. Romano and Andrew F. Siegel.

See Counterexample and Counterexamples in Probability and Statistics

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 on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.

See Counterexample and Counterexamples in Topology

Deductive reasoning

Deductive reasoning is the process of drawing valid inferences. Counterexample and Deductive reasoning are logic.

See Counterexample and Deductive reasoning

Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.

See Counterexample and Euclidean plane isometry

Euler's sum of powers conjecture

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem.

See Counterexample and Euler's sum of powers conjecture

Exception that proves the rule

"The exception that proves the rule" is a saying whose meaning is contested.

See Counterexample and Exception that proves the rule

Ganea conjecture

Ganea's conjecture is a now disproved claim in algebraic topology.

See Counterexample and Ganea conjecture

Generalization

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims.

See Counterexample and Generalization

Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See Counterexample and Geometry

Gorgias (dialogue)

Gorgias (Γοργίας) is a Socratic dialogue written by Plato around 380 BC.

See Counterexample and Gorgias (dialogue)

Hilbert's fourteenth problem

In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.

See Counterexample and Hilbert's fourteenth problem

Hypothesis

A hypothesis (hypotheses) is a proposed explanation for a phenomenon.

See Counterexample and Hypothesis

Imre Lakatos

Imre Lakatos (Lakatos Imre; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pre-axiomatic stages of development, and also for introducing the concept of the "research programme" in his methodology of scientific research programmes.

See Counterexample and Imre Lakatos

J. Arthur Seebach Jr.

See Counterexample and J. Arthur Seebach Jr.

James Franklin (philosopher)

James Franklin (born 1953) is an Australian philosopher, mathematician and historian of ideas.

See Counterexample and James Franklin (philosopher)

Logic

Logic is the study of correct reasoning.

See Counterexample and Logic

Loss function

In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event.

See Counterexample and Loss function

Lynn Steen

Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a professor of mathematics at St. Olaf College, Northfield, Minnesota, in the U.S. He wrote numerous books and articles on the teaching of mathematics.

See Counterexample and Lynn Steen

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. Counterexample and mathematical proof are mathematical terminology.

See Counterexample and Mathematical proof

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Counterexample and Mathematics

Minimal counterexample

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. Counterexample and minimal counterexample are mathematical terminology.

See Counterexample and Minimal counterexample

Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

See Counterexample and Natural number

Parity (mathematics)

In mathematics, parity is the property of an integer of whether it is even or odd.

See Counterexample and Parity (mathematics)

Pólya conjecture

In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors.

See Counterexample and Pólya conjecture

Philosophy

Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, value, mind, and language.

See Counterexample and Philosophy

Plato

Plato (Greek: Πλάτων), born Aristocles (Ἀριστοκλῆς; – 348 BC), was an ancient Greek philosopher of the Classical period who is considered a foundational thinker in Western philosophy and an innovator of the written dialogue and dialectic forms.

See Counterexample and Plato

Prima facie

Prima facie is a Latin expression meaning "at first sight", or "based on first impression".

See Counterexample and Prima facie

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

See Counterexample and Prime number

Proofs and Refutations

Proofs and Refutations: The Logic of Mathematical Discovery is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics.

See Counterexample and Proofs and Refutations

Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

See Counterexample and Rectangle

Rhombus

In plane Euclidean geometry, a rhombus (rhombi or rhombuses) is a quadrilateral whose four sides all have the same length.

See Counterexample and Rhombus

Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

See Counterexample and Rigour

Seifert conjecture

In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit.

See Counterexample and Seifert conjecture

Shape

A shape is a graphical representation of an object's form or its external boundary, outline, or external surface.

See Counterexample and Shape

Shear mapping

In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction.

See Counterexample and Shear mapping

Socrates

Socrates (– 399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and as among the first moral philosophers of the ethical tradition of thought.

See Counterexample and Socrates

Square

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles (90-degree angles, π/2 radian angles, or right angles).

See Counterexample and Square

Squeeze mapping

In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

See Counterexample and Squeeze mapping

State variable

A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system.

See Counterexample and State variable

Tait's conjecture

In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices".

See Counterexample and Tait's conjecture

Universal quantification

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any".

See Counterexample and Universal quantification

Witsenhausen's counterexample

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

See Counterexample and Witsenhausen's counterexample

References

[1] https://en.wikipedia.org/wiki/Counterexample

Also known as Counter example, Counter-example, Counterexamples, Proof by counterexample.

Counterexample, the Glossary

Table of Contents

See also

Interpretation (philosophy)

Methods of proof

References