The End 2016 Mathematics A To Z: Image


It’s another free-choice entry. I’ve got something that I can use to make my Friday easier.

Image.

So remember a while back I talked about what functions are? I described them the way modern mathematicians like. A function’s got three components to it. One is a set of things called the domain. Another is a set of things called the range. And there’s some rule linking things in the domain to things in the range. In shorthand we’ll write something like “f(x) = y”, where we know that x is in the domain and y is in the range. In a slightly more advanced mathematics class we’ll write f: x \rightarrow y . That maybe looks a little more computer-y. But I bet you can read that already: “f matches x to y”. Or maybe “f maps x to y”.

We have a couple ways to think about what ‘y’ is here. One is to say that ‘y’ is the image of ‘x’, under ‘f’. The language evokes camera trickery, or at least the way a trick lens might make us see something different. Pretend that the domain is something you could gaze at. If the domain is, say, some part of the real line, or a two-dimensional plane, or the like that’s not too hard to do. Then we can think of the rule part of ‘f’ as some distorting filter. When we look to where ‘x’ would be, we see the thing in the range we know as ‘y’.

At this point you probably imagine this is a pointless word to have. And that it’s backed up by a useless analogy. So it is. As far as I’ve gone this addresses a problem we don’t need to solve. If we want “the thing f matches x to” we can just say “f(x)”. Well, we write “f(x)”. We say “f of x”. Maybe “f at x”, or “f evaluated at x” if we want to emphasize ‘f’ more than ‘x’ or ‘f(x)’.

Where it gets useful is that we start looking at subsets. Bunches of points, not just one. Call ‘D’ some interesting-looking subset of the domain. What would it mean if we wrote the expression ‘f(D)’? Could we make that meaningful?

We do mean something by it. We mean what you might imagine by it. If you haven’t thought about what ‘f(D)’ might mean, take a moment — a short moment — and guess what it might. Don’t overthink it and you’ll have it right. I’ll put the answer just after this little bit so you can ponder.

Close up view of a Flemish Giant rabbit looking at you from the corner of his eye.
Our pet rabbit on the beach in Omena, Michigan back in July this year. Which is a small town on the Traverse Bay, which is just off Lake Michigan where … oh, you have Google Maps, you don’t need me. Anyway we wondered what he would make of vast expanses of water, considering he doesn’t like water what with being a rabbit and all that. And he watched it for a while and then shuffled his way in to where the waves come up and could wash over his front legs, making us wonder what kind of crazy rabbit he is, exactly.

So. ‘f(D)’ is a set. We make that set by taking, in turn, every single thing that’s in ‘D’. And find everything in the range that’s matched by ‘f’ to those things in ‘D’. Collect them all together. This set, ‘f(D)’, is “the image of D under f”.

We use images a lot when we’re studying how functions work. A function that maps a simple lump into a simple lump of about the same size is one thing. A function that maps a simple lump into a cloud of disparate particles is a very different thing. A function that describes how physical systems evolve will preserve the volume and some other properties of these lumps of space. But it might stretch out and twist around that space, which is how we discovered chaos.

Properly speaking, the range of a function ‘f’ is just the image of the whole domain under that ‘f’. But we’re not usually that careful about defining ranges. We’ll say something like ‘the domain and range are the sets of real numbers’ even though we only need the positive real numbers in the range. Well, it’s not like we’re paying for unnecessary range. Let me call the whole domain ‘X’, because I went and used ‘D’ earlier. Then the range, let me call that ‘Y’, would be ‘Y = f(X)’.

Images will turn up again. They’re a handy way to let us get at some useful ideas.

A Leap Day 2016 Mathematics A To Z: Surjective Map


Gaurish today gives me one more request for the Leap Day Mathematics A To Z. And it lets me step away from abstract algebra again, into the world of analysis and what makes functions work. It also hovers around some of my past talk about functions.

Surjective Map.

This request echoes one of the first terms from my Summer 2015 Mathematics A To Z. Then I’d spent some time on a bijection, or a bijective map. A surjective map is a less complicated concept. But if you understood bijective maps, you picked up surjective maps along the way.

By “map”, in this context, mathematicians don’t mean those diagrams that tell you where things are and how you might get there. Of course we don’t. By a “map” we mean that we have some rule that matches things in one set to things in another. If this sounds to you like what I’ve claimed a function is then you have a good ear. A mapping and a function are pretty much different names for one another. If there’s a difference in connotation I suppose it’s that a “mapping” makes a weaker suggestion that we’re necessarily talking about numbers.

(In some areas of mathematics, a mapping means a function with some extra properties, often some kind of continuity. Don’t worry about that. Someone will tell you when you’re doing mathematics deep enough to need this care. Mind, that person will tell you by way of a snarky follow-up comment picking on some minor point. It’s nothing personal. They just want you to appreciate that they’re very smart.)

So a function, or a mapping, has three parts. One is a set called the domain. One is a set called the range. And then there’s a rule matching things in the domain to things in the range. With functions we’re so used to the domain and range being the real numbers that we often forget to mention those parts. We go on thinking “the function” is just “the rule”. But the function is all three of these pieces.

A function has to match everything in the domain to something in the range. That’s by definition. There’s no unused scraps in the domain. If it looks like there is, that’s because were being sloppy in defining the domain. Or let’s be charitable. We assumed the reader understands the domain is only the set of things that make sense. And things make sense by being matched to something in the range.

Ah, but now, the range. The range could have unused bits in it. There’s nothing that inherently limits the range to “things matched by the rule to some thing in the domain”.

By now, then, you’ve probably spotted there have to be two kinds of functions. There’s one in which the whole range is used, and there’s ones in which it’s not. Good eye. This is exactly so.

If a function only uses part of the range, if it leaves out anything, even if it’s just a single value out of infinitely many, then the function is called an “into” mapping. If you like, it takes the domain and stuffs it into the range without filling the range.

Ah, but if a function uses every scrap of the range, with nothing left out, then we have an “onto” mapping. The whole of the domain gets sent onto the whole of the range. And this is also known as a “surjective” mapping. We get the term “surjective” from Nicolas Bourbaki. Bourbaki is/was the renowned 20th century mathematics art-collective group which did so much to place rigor and intuition-free bases into mathematics.

The term pairs up with the “injective” mapping. In this, the elements in the range match up with one and only one thing in the domain. So if you know the function’s rule, then if you know a thing in the range, you also know the one and only thing in the domain matched to that. If you don’t feel very French, you might call this sort of function one-to-one. That might be a better name for saying why this kind of function is interesting.

Not every function is injective. But then not every function is surjective either. But if a function is both injective and surjective — if it’s both one-to-one and onto — then we have a bijection. It’s a mapping that can represent the way a system changes and that we know how to undo. That’s pretty comforting stuff.

If we use a mapping to describe how a process changes a system, then knowing it’s a surjective map tells us something about the process. It tells us the process makes the system settle into a subset of all the possible states. That doesn’t mean the thing is stable — that little jolts get worn down. And it doesn’t mean that the thing is settling to a fixed state. But it is a piece of information suggesting that’s possible. This may not seem like a strong conclusion. But considering how little we know about the function it’s impressive to be able to say that much.