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?

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My (non-mathematician) wife and I had a chat about that question! I can’t claim to have noticed the error as I gave all three possible answers (injection, bijection, direction). BTW, I used to sit near you on Level 12, back in PhD days.

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Hi!

(But wasn’t it Level 11?)

By coincidence my (former!) PhD supervisor from Leeds, Garth Dales, is visiting Nottingham this week.

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Yes, you’re right, level 11 it was. I was first on the left after the big fluid dynamics machine. I believe there was a fabled level 12 in the Earth Sciences department, or somewhere.

I was stats, 1988 – 1993 (not the fastest PhD), supervisor Prof JT Kent.

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I guess the key phrase here is `at most’; the question’s author probably meant to exclude inverse images of singletons which had size 2, 3, or more. In doing so they unwittingly allowed empty inv. images of singletons!

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That is certainly one of the things that could have happened.

However, in my classes (with slightly more careful use of quantifiers) I do actually use “at most one” as one way to define injections, “at least one” as a way to define surjections and “exactly one” as a way to define bijections.

In fact you can see exactly what I told our first-year students this year in the video (

link now corrected) atwith the definition of injective at (

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First of all I don’t like the way the way the question is phrased because it sounds like “there is at most one element in the domain that is mapped to every element of the codomain”, i.e. either the function is almost arbitrary (there are some slight and bizarre restrictions) or the domain and codomain consist of exactly one element each.

But, if we assume a more sensible reading such as “the function associates at most one element of the domain with any one element of the codomain, and every element of the codomain is associated in this way”, then it becomes clear why “bijection” is the answer they were looking for.

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By saying “at most one”, I think that they must surely be allowing the possibility of zero? (If you want “exactly one” you need to say so?)

I agree that there is also an ambiguity/inaccuracy of the type you mention, but it is the “at most one” that I object to most strongly.

The use of “each and every” is a bit odd. I don’t think you can go from “every element of the codomain is associated with at most one …” to obtain “every element of the codomain is associated with at least one …”, so perhaps you are being overly generous to them? If they meant what you suggest, I think they would have needed to clarify it further.

Meanwhile, I have corrected the links to my YouTube videos, which weren’t working properly. Instead of linking to the video of Lecture 15, they were taking people to the start of Lecture 1. Sorry! It looks like there are some pitfalls when trying to link to a video which is part of a longer playlist.

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