Foundations of Pure Mathematics 2018-19 edition

The links I previously published to the old Echo360 archives of my module Foundations of Pure Mathematics are no longer working, but I have now attempted to make public the Echo360 recordings from 2018-19 at

https://echo360.org.uk/section/4c490e8b-27f7-4fb8-b97f-251bad1ed2ad/public

so hopefully those recordings are now available.

Negations of statements involving implications

I have just sent the following message to my first-year students. But I am not sure whether they will find this helpful or confusing.

Since they have a Class Test later today, this may have been the wrong time to send it anyway!

Message sent to first-year students follows:

Looking through some more of your work on Practice Coursework 1, I have noticed one more logical reasoning point that has come up, related to negating statements (when trying to prove things by contradiction). This can be a bit tricky if you are trying to negate a statement which involves an implication sign or the word “if”. (In the solutions you will see that I have generally been negating statements of a more tractable type. I recommend that, if you can, you keep things simple when you are negating things!)

In order to prove true a statement of the form A implies B, some of you have said “Assume towards a contradiction that A implies that (B is false)” (or something like that). However, the correct negation of “A implies B” is instead “A does not imply B”.

One comment here is that it should not be possible for a statement and its negation to both be false. Nor should it be possible for them both to be true.

In the below, we assume that n is an integer, but make no other assumptions about n before considering whether certain logical implications are true or false.

Consider the false statement (about integers n) that says

“n is divisible by 3 implies that n is divisible by 6”.

(By false here we really mean “false for at least one integer n”. So consider n=3.)

Here the statement

“n is divisible by 3 implies that n is not divisible by 6″

is also false! (Again meaning false for at least one n. This time consider n=6.)

The correct negation of the first false statement is

“n is divisible by 3 does not imply that n is divisible by 6”

(by which, this time, we mean that there is at least one integer n such that n is divisible by 3 but n is not divisible by 6).

As you would expect, this correct negation of the first false statement is actually true.

Note that we have followed certain conventions here when discussing whether these implications are true or false. You may wish to rewrite the various claims above more formally using “For all integers n” and “There exists an integer n”  to clarify things.

Using the mathematicians’ truth value for “implies”, you may wish to note some relatively strange things:

“4 is divisible by 3 implies that 4 is divisible by 6”

is (officially) true, and the statement

“4 is divisible by 3 implies that 4 is not divisible by 6″

is also (officially) true! So that can’t be the correct negation of the first statement.

But

“4 is divisible by 3 does not imply that 4 is divisible by 6”

is false!

Reading list for Functional Analysis

I have been asked on YouTube about the reading list for my old module G14FUN Functional Analysis. It looks like that is not included in the materials available on UNoW at https://rdmc.nottingham.ac.uk/handle/internal/257 so what I’ll do is I’ll quote the reading list here. To be precise, this is my current reading list for the module Further Topics in Analysis, but the syllabus is very similar. (The main difference is that Further Topics in Analysis includes some material on measure theory, and a slight reduction in the functional analysis content).

Books

The following books are all well worth looking at. Although no single book is ideal for the module, the book of Allan is probably the closest for the material on functional analysis (although the more advanced material in that book goes well beyond the scope of this module). Indeed, I first learned most of the material in this module from the lectures of Allan (on which his book is based). The books by Jameson and Bollobás and Rudin’s book on Real and Complex Analysis are also highly recommended.  In particular, the early chapters of the latter book are a very good source for the material on measure theory in this module.

  • Allan, Graham R. (Prepared for publication by H. Garth Dales). Introduction to Banach Spaces and Algebras. Oxford Graduate Texts in Mathematics, 2011.
  • Bollobás, Béla. Linear analysis : an introductory course / Béla Bollobás, 2nd ed. Cambridge University Press, 1999.
  • Halmos, Paul R. Finite-dimensional vector spaces / Paul R. Halmos, 2nd ed. Springer, 1987.
  • Jameson, G. J. O. Topology and normed spaces / G. J. O. Jameson. Chapman and Hall, 1974.
  • Pedersen, Gert K. Analysis now / Gert K. Pedersen. Springer, 1989.
  • Rudin, Walter. Functional analysis / Walter Rudin, 2nd ed. McGraw-Hill, 1991.
  • Rudin, Walter. Real and complex analysis / Walter Rudin, 3rd ed. McGraw-Hill, 1987.
  • Simmons, George Finlay. Introduction to topology and modern analysis / George Finlay Simmons. Krieger, 2003.

 

Notation for composite functions

From seeing my children’s school work, it looks as if GCSE maths in the UK uses what looks to me to be confusing notation for composite functions. Essentially they write fg where I would write f \circ g. For an example, see the BBC web page

https://www.bbc.com/bitesize/guides/z36vcj6/revision/6

To be precise, they write things like fg(4)=f(g(4)).

Surely this can cause confusion with the pointwise product function fg given by x \mapsto f(x) g(x)?

I suppose that f \circ g might look weird and frightening?

Another name for powers of 2?

I sometimes run a session for school children (typically aged 10-12) about the grains of rice on a chessboard, and the Tower of Hanoi puzzle. Here powers of two are relevant, but I wanted a child-friendly name for these numbers, so I have been calling them “doubling numbers”.  At the end of the session I have provided a sheet with the first 65 doubling numbers (starting from 1). But I’m no longer entirely happy with calling them “doubling numbers”.

I don’t want to call them powers of two unless I have to. But “numbers you get by starting from 1 and doubling again and again” is a bit unwieldy.

Maybe I will stick to “powers of two” in the end.

Nottingham Festival of Science and Curiosity

This year’s Festival of Science and Curiosity runs from 13-20 February 2019.

The University of Nottingham’s School of Mathematical Sciences is contributing various activities on Saturday 16th February.

I’ll be involved in the activities at Green’s Windmill and Science Centre. We will have a variety of mathematical games, puzzles and exhibits, including Rubik’s Cubes, Towers of Hanoi, and gyroscopes balancing on strings.

Separately, at a stall in Broadmarsh Shopping Centre, there will be a group of PhD students from our MASS group (Modelling and Analytics for a Sustainable Society). They have some activities and games relating to mathematical research concerning sustainability and antimicrobial resistance.

Why do we need a property called surjectivity?

Here is a question I am often asked by my first-year students. Why do we need a name for the condition of surjectivity, when you can always change the codomain to be equal to the image, and make your function surjective that way? (I am avoiding the word ‘range’, because that turns out to be used differently by different authors.)

I don’t know currently what the best answer to this is, though I have some ideas.

I could refer to mathematical tradition, and talk about the flexibility of the standard approach. For example, I think it is quite useful that we have a large set of functions from \mathbb{R} to \mathbb{R} given by polynomial functions with real coefficients, and it would be inconvenient if we couldn’t say where these functions were mapping to without calculating the image. Off the top of my head, consider, for example, a polynomial function x \mapsto x^6-3x^3 +x^2 defined on \mathbb{R}. What is its image? Do we really want to have to calculate the image before we can say what the codomain is? It seems relatively easy just to treat it as a function from \mathbb{R} to \mathbb{R} that isn’t surjective.

I suppose though that we could just say that it is a function from \mathbb{R} to its image. Is there any problem with that? Well maybe it gets a bit complicated later when you start looking at homomorphisms in algebra, but you can probably work round that.

I expect that a category theorist would have something to say on this issue! But what is the best thing to say to a first-year undergraduate?