Counterexample in a sentence

However, the failure to find a counterexample after extensive search does not constitute a proof that no counterexample exists nor that the conjecture is true, because the conjecture might be false but with a very large minimal counterexample.

A casual verifier of the counterexample may not think to change the colors of these regions, so that the counterexample will appear as though it is valid.

In the case of the latter, the first counterexample found involved numbers in the millions, although subsequently it has been found that the minimal counterexample is smaller than that.

Also, if my counterexample doesn't do it for you (because you can't understand it.

As much as we like to believe in evolution, Florida is a pretty compelling counterexample.

Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps.

Background Euler had an equality for four fourth powers ; this however is not a counterexample because no term is isolated on one side of the equation.

Examples One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion.

For more than two events, a mutually independent set of events is (by definition) pairwise independent; but the converse is not necessarily true (see below for a counterexample).

In 1988, Roger Frye found the smallest possible counterexample :: for by a direct computer search using techniques suggested by Elkies.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

In the domain of field research, the Pirahã language is claimed to be a counterexample to the basic tenets of universal grammar.

Logically, no number of positive outcomes at the level of experimental testing can confirm a scientific theory, but a single counterexample is logically decisive: it shows the theory, from which the implication is derived, to be false.

Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done.

Non-finitely generated abelian groups Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.

Note that infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.

Such terms, it may be argued, are always intensional since they connote the property 'meaningless term' but this paradox does not constitute a counterexample to the claim that without intension a word has no meaning.

The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.

This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found.

Thus, if one could prove that neither of these types of counterexample could exist, then all positive integers would have a trajectory that reaches the trivial cycle.