Injections vs bijections: University Challenge error

My colleague David Hodge just showed me an extract from a recent University Challenge where they were asked the following question.

“Which type of function associates at most one element of the domain with each and every element of the codomain?”

The students weren’t sure, but (in my opinion!) correctly answered “Injection”. However they were told that they were wrong, and that the correct answer was “Bijection”.

Maybe we should write in and complain?


Confusion involving less than or equal

Every year a number of my first-year students are confused by the “less than or equals” relation.

Some students are confused when I claim that 2 \leq 3 or when I claim that 1 \leq 1 (or both).

You can see what happened this year when I mentioned that 2 \leq 3:

As I said, I get questions about this every year.

I think that the confusion goes via a subtle conversion of “or” to “and”, as in  “it might be less than and it might be equal”, and this jars when they see examples where it is blatantly one and not the other.

Some students like diagrams a lot, and so number line sketches could help illustrate the concept more clearly.

More YouTube auto-generated collections

Is there a list of YouTube auto-generated collections?

I have just found what they call #BanachSpace at

Aha, and I see that you can search for #FunctionalAnalysis and find
and #MathematicalAnalysis finds

I suppose they have a lot of these auto-generated lists.

Indeed there is #Mathematics at

My own videos aren’t that visible on #Mathematics: I see my Introduction to Foundations of Pure Mathematics is currently 30th in the list at

On the other hand, I can’t complain about being one place ahead of Simon Singh’s excellent talk at Google (associated with his book) about some of the mathematics hiding in The Simpsons.

YouTube vs iTunes U

As an experiment, I just did a search for “Equivalence Relations” (with or without the s) on iTunes. Very little turns up in the search results. The only Collection shown is my G11FPM, and the only iTunes U Episode shown is one of my lectures. That seems surprisingly little given how many institutions are making materials available out there.

Compare this with a search on YouTube:

gives over 3000 results, and my  lecture on this only just makes it into the current top 20.

Is it just this topic where few of the iTunes U Universities are contributing? Or is it a more general pure mathematics shortage on iTunes U? Or is it a metadata problem, where not enough content detail is included in the iTunes descriptions?

Can undergraduates read my research papers?

Can undergraduates read my research papers?

Well, most of them are available electronically from various Open Access repositories, and also from (at least for as long as we are allowed personal web pages). But that’s not exactly what I meant!

What I really mean is, can undergraduates get anything out of reading my research papers? I would like to think that some of them have enough accessible material in that the answer is yes.

If you are an undergraduate thinking of looking at some of my papers, I might suggest the following pair of papers as particularly accessible. The first does need some knowledge of measure theory, but most of the second is about elementary real analysis (differentiable and Lipschitz functions of a real variable).

  • Convergence from below suffices, Irish Mathematical Society Bulletin, 59 (2007), 65-70: pdf
  • (With J. Craig and P. Patrick) Removability of exceptional sets for differentiable and Lipschitz functions, to appear in Contemp. Math., Proceedings of the Seventh Conference on Function Spaces, Edwardsville, Illinois (2014): pdf

Maybe also

  • (With S. Morley and H. Yang) Swiss cheeses and their applications, to appear in Contemp. Math., Proceedings of the Seventh Conference on Function Spaces, Edwardsville, Illinois (2014): pdf

for those who want to know what my Swiss cheese icon is all about!

Uta Feinstein’s Art (OT)

I have helped my wife (Uta Feinstein) set up a personal Artist’s web page at and we are gradually uploading selected work to her Gallery

New most popular pure maths video on YouTube

I really don’t know how YouTube measures popularity in its auto-generated lists.

Looking at

I see that  my lecture on “Sets and subsets” is now listed first in the Popular Pure Mathematics Videos there … currently it has 10 views!

This module is now represented twice in the “Popular Pure Mathematics Playlists” because another YouTube user has made a small playlist with some of the videos in, and his version is also apparently very popular, as measured by YouTube.

Meanwhile, the first lecture (well, actually the version from 2012-13, as explained earlier) is actually beginning to pick up hits faster now … so of course it has lost top spot and has moved down the list of popular videos.

Anyway, thanks again to Sally Hanford, the first 12 videos from G11FPM Foundations of Pure Mathematics are now on YouTube Edu and on iTunes U.