When teaching mathematics, one is often faced with the task of convincing a student that the converse of a given theorem is false or that the theorem is as strong as possible. The usual method involves the production of a counterexample. However, ever a quick glance at a wide range of textbooks shows a singular lack of variety in the counterexamples exhibited, arising no doubt from a lack of imagination on the part of the authors. Indeed, an inexperienced student might be led to conjecture that the following statements are theorems:

1. | The function | ||||||||||

2. | The real interval [0, 1] is the only uncountable set. | ||||||||||

3. | The function defined on [0, 1] by
is the only function that is not Riemann integrable. | ||||||||||

4. | The only noncommutative operations in algebra are multiplication of | ||||||||||

5. |
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6. |
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7. | The numbers π and | ||||||||||

8. | Cubic and quartic polynomials are soluble by radicals, but nobody knows the details. | ||||||||||

9. | √2 is the only irrational number. | ||||||||||

10. | The alternating group on four symbols is the only finite group that does not satisfy the converse of Lagrange's Theorem. | ||||||||||

11. | Any result about the natural numbers can be proved by induction, but the details can always be omitted. | ||||||||||

12. | The only Pythagorean triple is (3, 4, 5). |