Disproving a Conjecture by the Use of a Counter Example

Disproving a Conjecture with 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.

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.

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.

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.