The predictability of counterexamples

Desmond MacHale
Department of Mathematics, University College, Cork, Ireland

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  f (x) = |x| is the only real function that is continuous but not differentiable.

2.

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

3.

The function defined on [0, 1] by

f (x) =
 ì í î
 0,   x rational 1,   x irrational

is the only function that is not Riemann integrable.

4.

The only noncommutative operations in algebra are multiplication of 2×2 matrices, subtraction of integers, and composition of permutations on three objects.

5.

 ∞ ∑ 1 n is the only divergent series. n=1

6.

 ∞ ∑ (–1)n n is the only series that is convergent but not absolutely convergent. n=1

7.

The numbers π and e are transcendental, but the proof is beyond the scope of any textbook.

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).