What Is Counterexample In Math

Exception to a proposed general rule

A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does and then rigorously in the fields of mathematics and philosophy.^[i] For instance, the fact that "John Smith is not a lazy student" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."^[2]

In mathematics, the term "counterexample" is as well used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done past because a instance where a part of the hypothesis is non satisfied and the conclusion of the theorem does not hold.^{[ citation needed ]}

In mathematics [edit]

In mathematics, counterexamples are oftentimes used to prove the boundaries of possible theorems. Past using counterexamples to show that sure conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to change conjectures to produce provable theorems. Information technology is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.^[iii]

Rectangle example [edit]

Suppose that a mathematician is studying geometry and shapes, and she wishes to testify certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false.

In this instance, she can either effort to bear witness the truth of the argument using deductive reasoning, or she tin can attempt to detect a counterexample of the statement if she suspects it to be fake. In the latter case, a counterexample would exist a rectangle that is not a square, such as a rectangle with two sides of length 5 and 2 sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did notice had four sides. She then makes the new conjecture "All rectangles accept four sides". This is logically weaker than her original conjecture, since every square has four sides, only not every four-sided shape is a square.

The above instance explained — in a simplified mode — how a mathematician might weaken her conjecture in the face of counterexamples, just counterexamples can also exist used to demonstrate the necessity of certain assumptions and hypothesis. For instance, suppose that subsequently a while, the mathematician to a higher place settled on the new conjecture "All shapes that are rectangles and have 4 sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must take 'four sides of equal length'. The mathematician and then would like to know if she can remove either assumption, and even so maintain the truth of her conjecture. This means that she needs to check the truth of the following two statements:

"All shapes that are rectangles are squares."
"All shapes that have 4 sides of equal length are squares".

A counterexample to (ane) was already given above, and a counterexample to (two) is a non-square rhomb. Thus, the mathematician now knows that both assumptions were indeed necessary.

Other mathematical examples [edit]

A counterexample to the statement "all prime numbers are odd numbers" is the number ii, as it is a prime number merely is not an odd number.^[1] Neither of the numbers vii or ten is a counterexample, as neither of them are enough to contradict the statement. In this instance, 2 is in fact the only possible counterexample to the argument, even though that lonely is enough to contradict the statement. In a similar manner, the argument "All natural numbers are either prime or composite" has the number one equally a counterexample, as i is neither prime number nor composite.

Euler's sum of powers conjecture was disproved by counterexample. It asserted that at to the lowest degree n n ^th powers were necessary to sum to another n ^th power. This conjecture was disproved in 1966,^[4] with a counterexample involving north = v; other n = five counterexamples are now known, as well as some northward = 4 counterexamples.^[5]

Witsenhausen'southward counterexample shows that it is not always true (for control bug) that a quadratic loss function and a linear equation of evolution of the country variable imply optimal control laws that are linear.

Other examples include the disproofs of the Seifert theorize, the Pólya theorize, the theorize of Hilbert's fourteenth problem, Tait's theorize, and the Ganea theorize.

In philosophy [edit]

In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not utilize in certain cases. Alternatively, the kickoff philosopher tin can alter their claim and so that the counterexample no longer applies; this is analogous to when a mathematician modifies a theorize because of a counterexample.

For example, in Plato'due south Gorgias, Callicles, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better.

Simply Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied course of nobles, fifty-fifty though the masses are prima facie of worse graphic symbol. Thus Socrates has proposed a counterexample to Callicles' claim, by looking in an surface area that Callicles maybe did not expect — groups of people rather than individual persons.

Callicles might challenge Socrates' counterexample, arguing maybe that the common rabble really are better than the nobles, or that even in their big numbers, they notwithstanding are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim, or change it and then that the counterexample no longer applies. For example, he might modify his claim to refer just to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob.

As it happens, he modifies his claim to say "wiser" instead of "stronger", arguing that no amount of numerical superiority can make people wiser.

See besides [edit]

Contradiction
Exception that proves the dominion
Minimal counterexample

References [edit]

^ ^a ^b "Mathwords: Counterexample". www.mathwords.com . Retrieved 2019-eleven-28 .
^ Weisstein, Eric W. "Counterexample". mathworld.wolfram.com . Retrieved 2019-11-28 .
^ "What Is Counterexample?". www.cutting-the-knot.org . Retrieved 2019-eleven-28 .
^ Lander, Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers" (PDF). Bulletin of the American Mathematical Society. Americal Mathematical Social club. 72 (6): 1079. doi:10.1090/s0002-9904-1966-11654-3. ISSN 0273-0979. Retrieved 2 Baronial 2018.
^ Elkies, Noam (Oct 1988). "On A4 + B4 + C4 = D4" (PDF). Mathematics of Computation. 51 (184): 825–835.