14. Write What can be done to disprove a conjecture? Explain.

To disprove a conjecture, one must provide a counterexample. A counterexample is a specific instance or case that demonstrates that the conjecture is not universally true. Here are the steps to effectively disprove a conjecture: 1. **Understand the Conjecture**: Clearly define the conjecture and understand its claims. This includes identifying the conditions under which the conjecture is said to hold true. 2. **Identify the Domain**: Determine the domain or set of elements that the conjecture applies to. This helps in knowing where to look for potential counterexamples. 3. **Search for Counterexamples**: Look for specific cases or examples that contradict the conjecture. This could involve: - Testing various values or scenarios that fit within the conjecture's parameters. - Using logical reasoning to find instances where the conjecture fails. 4. **Verify the Counterexample**: Once a potential counterexample is found, verify that it indeed contradicts the conjecture. Ensure that it meets the conditions of the conjecture but leads to a different conclusion. 5. **Document the Findings**: Clearly present the counterexample, explaining how it disproves the conjecture. This should include a detailed explanation of why the example is valid and how it contradicts the original claim. 6. **Consider Edge Cases**: Sometimes, conjectures may hold true for most cases but fail in edge cases. Exploring these can be crucial in finding a counterexample. By following these steps, one can effectively disprove a conjecture and contribute to the understanding of the mathematical or logical principles involved.