How to Master the World of Conjectures and Counterexamples

A conjecture is a statement you believe true from a pattern but have not proved. A single counterexample disproves it outright, which is why finding one is often faster than attempting a proof.

• They refine and correct conjectures.
• They prevent mathematicians from pursuing false statements.
• They offer clarity on the limitations of a statement’s accuracy.

Worked examples: conjectures and counterexamples

Practice questions on conjectures and counterexamples

1. Conjecture: All positive integers are greater than (0). Is this true or false? If false, provide a counterexample.
2. Conjecture: The square of any integer is positive. Is this true or false? If false, provide a counterexample.
3. Conjecture: All birds can fly. Is this true or false? If false, provide a counterexample.

1. True. By definition, positive integers are greater than (0).
2. True. The square of any integer, whether positive or negative, is always positive.
3. False. Counterexample: Ostriches, penguins, and kiwis are birds that cannot fly.

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