Disproving a Conjecture by the Use of a Counter Example

Disproving a Conjecture by the Use of a Counter Example

Disproving a Conjecture with a Counterexample

Defining a Conjecture and a Counterexample

  • A conjecture in mathematics is a statement believed to be true based on observations.
  • A counterexample is a specific case or example that contradicts a conjecture’s statement. It’s a powerful tool in mathematical reasoning and problem-solving.

Principle Behind Using a Counterexample

  • Establishing the truth of a mathematical statement requires a rigorous and deductive proof.
  • However, proving a statement is false needs just one instance where it doesn’t hold; this is where a counterexample comes to play.

Process of Finding and Using a Counterexample

  • Firstly, understand the conjecture well, be aware of its boundaries and its expressed conditions.
  • Secondly, imagine opposing scenarios. Assume the conjecture to be false and think about what that world looks like.
  • Use the conditions of the conjecture to generate potential counterexamples. In some cases, this might require trial and error, but in others, you might spot a ‘gap’ in the conjecture that you can aim to exploit with a counterexample.
  • Always check your counterexample. Does it meet all of the conditions for use in the conjecture? Does it demonstrate that the conjecture doesn’t hold?
  • A valid counterexample immediately disproves the conjecture, regardless of how many other cases the conjecture is true for.

Tips for Using Counterexamples

  • Counterexamples might not always be intuitive. You might need to think creatively and challenge your own preconceptions of the problem.
  • A great counterexample is one that not just disproves a conjecture but offers insight into why the conjecture fails and potentially how it could be modified to be correct.
  • Work on developing the skill to quickly generate potential counterexamples as it is a highly useful tool in both problem solving and in mathematical proofs.