What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.

In many branches of mathematics, it seems to me that a good counterexample can be worth just as much as a good theorem or lemma. The only branch where I think this is explicitly recognized in the literature is topology, where for example Munkres is careful to point out and discuss his favorite counterexamples in his book, and Counterexamples in Topology is quite famous. The art of coming up with counterexamples, especially minimal counterexamples, is in my mind an important one to cultivate, and perhaps it is not emphasized enough these days.

So: what are your favorite examples of counterexamples that really illuminate something about some aspect of a subject?

Bonus points if the counterexample is minimal in some sense, bonus points if you can make this sense rigorous, and extra bonus points if the counterexample was important enough to impact yours or someone else's research, especially if it was simple enough to present in an undergraduate textbook.

• @Ben Linowitz. It is in the small and delightful book, J. W. S. Cassels, Lectures on elliptic curves. – Regenbogen Mar 2 '10 at 14:39
• @Ben Linowitz. Oh I am sorry for saying irrelevant things. I must confess I do not know anything at all. Maybe the following MSRI video might interest you(if you were not already aware of it)... msri.org/communications/vmath/VMathVideos/VideoInfo/3821/… – Regenbogen Mar 2 '10 at 17:38
• From a pedagogical standpoint, sometimes the minimal counterexample isn't the best one; in particular if it is "too small" to exhibit important general features of what's going on. – benblumsmith Jul 22 '11 at 12:55

I am quite keen of a counterexample in Strichartz estimates by Thomas Wolff that I think satisfies the requirements in your question. The problem was whether or not there are $L^p$ Strichartz estimates that lose no derivatives. Wolff gave a counterexample that, assuming the best control possible, then this is not possible for $p>2$. It uses the result by Charles Fefferman that the disk multiplier $S_1$ is not bounded on $L^p,~p\neq 2$. This is brilliantly explained by Professor Tao in the Notice of the AMS article "From Rotating Needles to Stability of Waves: Emerging Connections Between Combinatorics, Analysis, and PDE". It goes (more-or-less) like this:

Let $B(0,1)$ denote the unit ball of centre $0$ and radius $1$. Wolff showed that the inequality \begin{equation*} ||u||_{L^p([1,2]\times \mathbb{R}^n)}\leq C||f||_{L^{\infty}(B(0,1))} \end{equation*} fails, for $f$ bounded on $B(0,1)$.

Wolff used a mixture of Kakeya tube constructions, and the manipulation of specially constructed waves consisting of bump and exponential functions adapted to the tubes.

Another one of my favorite counter examples is $2\mathbb{Z}$ which is a RNG, or a ring without identity.

In the spirit of "examples do not prove anything in full generality (even if you have billions of them)", here is my favorite one: $$gcd(n^{17}+9,(n+1)^{17}+9)=1$$ is true for all integers $$1\leq n<8424432925592889329288 197322308900672459420460792433$$, but false for $$n=8424432925592889329288 197322308900672459420460792433$$.

I like "the deleted Tychonov plank" which is described in "Counterexamples in Topology".

This space provide us a pure algebraic counterexample:

A commutative ring is called a $Z$-ring if all its elements are zero divisors. At first glance, it seems that an extension of a $Z$-ring by a $Z$-ring is necessarily a $Z$ ring. But this space provides a counterexample. The reason is explained here.

http://arxiv.org/abs/1307.5836

To be honest, I spent a few weeks to give a (positive) proof for this ring-extension statment, but finaly I found this counterexample in the book "Counterexamples in Topology" which excited me a lot.

As a consequence of this post we ask:

Are there two $Z$-rings $R_{1}$ and $R_{2}$ such that for every short ring exact sequence $$0\to R_{1} \to S \to R_{2} \to 0$$ $S$ must be a $Z$-ring?

One might expect that for any function $f$ in the Hilbert space $L^2$ with orthonormal basis $\{ f_i \}$, the generalized Taylor series $$\sum_{j=1}^\infty \langle f, f_j \rangle f_j$$ converges pointwise to $f$ almost everywhere. Carleson's theorem gives that this is true for the standard Fourier basis of $L^2([0,1])$, but there exist uncountably many pairs of functions $g$ and orthonormal (ordered) bases $\{ f_i \}$ such that the generalized Fourier series above diverges pointwise almost everywhere, as discussed here.

Square root and square power are not inverse operations

In fact, $$\sqrt{x^2}=|{x}|$$, which is defined for every $$x\in\mathbb{R}$$, while $$({\sqrt{x}})^2=x$$ is defined only for $$x\geq 0$$.

The tensor product over a ring is not left exact

Consider the short exact sequence $$0\to2\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}_2\to0$$ and tensor it by $$\mathbb{Z}_2$$ over $$\mathbb{Z}$$. It is well-known that this produces the exact sequence $$2\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}_2 \to \mathbb{Z}_2 \to \mathbb{Z}_2\otimes_{\mathbb{Z}}\mathbb{Z}_2 \to 0$$ where the first arrow is the $$0$$ map even if $$2\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}_2$$ is non-zero (in particular, it cannot be injective).

In statistics not all distributions worth studying have mean, variance, skewness, or higher moments. Thus applications of the Central Limit Theorem may break down in such cases.

For example we have the Cauchy distribution:

$$f(x; x_0,\gamma) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} = { 1 \over \pi \gamma } \left[ { \gamma^2 \over (x - x_0)^2 + \gamma^2 } \right],$$

with location parameter $$x_0$$ (=median/mode) and scale parameter $$\gamma$$ (=half width at half maximum).

The Cauchy distribution is also the distribution of the ratio of two variables $$U$$ and $$V$$, both of which are $$\sim N(0,1)$$. The moments do not converge.

Image from Wikipedia: the example which shows that exp(zw) is not equal to exp (exp(z)^ w) the carrot sign means raised to the power another one a continuous function of a complex variable need not have primitive in a region.the example is f(z) = square ( | z| ).

• why -2c marks these are simple but important counter examples – Anil P Jul 1 '11 at 16:53
• What does exp(exp(z), w) mean? – LSpice May 13 '16 at 4:56

protected by Qiaochu YuanSep 21 '17 at 7:05

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