The Role of Counterexamples in Disproving Mathematical Claims

counterexample

an example that proves that a conjecture or statement is false

A counterexample is a specific example or case that disproves a statement or claim. In mathematics, when trying to prove a general statement or conjecture, one common way to do so is by providing counterexamples. A counterexample essentially shows that the statement is not universally true.

For example, suppose you have a conjecture which states, “All even numbers are divisible by 4.” To disprove this statement, you would need to find a single even number that is not divisible by 4. Let’s consider the number 6. It is an even number, but it is not divisible by 4 (since 6 divided by 4 leaves a remainder of 2). Hence, 6 serves as a counterexample to the conjecture.

By providing a counterexample, you demonstrate that the proposed statement is false, as it does not hold true for all cases. Counterexamples play an important role in a mathematical argument by challenging hypotheses and forcing mathematicians to refine their theories or reevaluate their assumptions.

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