Understanding Counterexamples in Mathematics: Disproving Statements and Conjectures

Counterexample

A counterexample is a specific example that disproves a statement or conjecture

A counterexample is a specific example that disproves a statement or conjecture. It is used to show that a general statement is not always true by providing a specific case where it fails.

Let’s consider an example to illustrate this concept:

Statement: “All prime numbers are odd.”

To provide a counterexample, we need to find a prime number that is not odd.

Counterexample: The number 2 is a prime number, but it is even (not odd). Therefore, the statement “All prime numbers are odd” is disproved by the counterexample of the number 2.

By providing a specific case where the prime number is not odd, we have shown that the statement is not universally true. Hence, the statement cannot be considered a valid generalization.

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