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?

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My open educational resources

As well as my videos on YouTube and iTunes, some of my previous University of Nottingham modules are available in full (including handouts etc.) on U-Now. Apparently these modules are now also available on OER commons.

 

 

Moebius strips

Yesterday was a first-year class where I briefly looked at the quotient mapping associated with an equivalence relation, and then showed the class a cylinder and a Moebius strip. That gave me the chance to ask them “Why did the chicken cross the Moebius strip?” Someone always gives me the “correct” answer, as long as I am patient and encouraging!

Surface pro scare

When I detached the cover from my surface pro today at the start of my class I was faced with a blank screen, and multiple (and increasingly desperate) long presses of the power button had no effect. I thought I was going to have to do the best I could with the resident desktop PC instead. But first I thought I’d try attaching the cover again and then opening it. Amazingly the machine sprang back to life!

The mysteries of computers …. (phew!)

Base k vs Modulo k

Below is a message I just sent to our first year students about a connection between modular arithmetic and working in other bases. I don’t know if this helps, but somehow arithmetic modulo 10 always seems so easy to explain!

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In our introduction to modular arithmetic, I suggested a connection between “modulo k” and “base k”. For positive integers, when  working modulo 10, using decimal notation, we only care about the last decimal digit. (That is the same as the remainder when you divide by 10.)

If you work in other bases, the connection is similar: base 8 (octal) is not the same thing as modulo 8, but for positive integers, the remainder when you divide by 8 is the same as the last octal digit. For example 25 in base 10 is congruent to 1 modulo 8, and 25 base 10 is written as 31 in octal. Any power of 31 (octal) still ends in 1 (octal). Returning to decimal notation, 25^n is congruent to 1 (mod 8) for all positive integers n, and so 25^n-1 is always divisible by 8.

Exercise: for which positive integer values of n is 6^n+1 divisible by 7?

 

Drawboard PDF and Pen Attention

Long-time readers will know that I use Pen Attention by Kenrick Mock to highlight the cursor position when I want a digital pointer in my classes.

I found out today that when Drawboard PDF is in its (most) fullscreen mode, the Pen Attention highlighter is invisible. So I will have to avoid that if I want to do any digital pointing! I can still have plenty of writing space, just not everything.

Drawboard PDF

As I haven’t managed to resolve some of the inking issues when using Bluebeam PDF Revu on my Surface Pro 4, I am going to give Drawboard PDF a try when annotating my lecture/workshop slides. The inking is very good! However it takes quite a few clicks to insert a blank page if needed, so it will be best if I have the right number of blank pages in my document before I start. Actually, one of the most common requests from students last year was that I should include enough blank space in their notes for them to write down my annotations, so that gives me two reasons to do so. I’ll base the number of blank pages on the number of extra pages I needed last year.