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.
More Answers:
Exploring Conjectures: The Unproven Hypotheses Guiding Mathematical DiscoveriesThe Power of the Contrapositive: Strengthening Arguments in Mathematics
Exploring the Converse of Conditional Statements in Mathematics: Understanding the Switched Hypothesis and Conclusion in Logical Reasoning and Theorems