How to Master the World of Conjectures and Counterexamples
- They refine and correct conjectures.
- They prevent mathematicians from pursuing false statements.
- They offer clarity on the limitations of a statement’s accuracy.
Examples
Practice Questions:
- Conjecture: All positive integers are greater than \(0\). Is this true or false? If false, provide a counterexample.
- Conjecture: The square of any integer is positive. Is this true or false? If false, provide a counterexample.
- Conjecture: All birds can fly. Is this true or false? If false, provide a counterexample.
- True. By definition, positive integers are greater than \(0\).
- True. The square of any integer, whether positive or negative, is always positive.
- False. Counterexample: Ostriches, penguins, and kiwis are birds that cannot fly.
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