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/ Mathematical question - can you map every x to every y?

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A year 10 student asked me a question today,"can you have a function that maps every x value to every y value?". When I dug into what he was asking, he wanted to plot a graph which shaded in the entire (x,y) plane.

A few seconds thought and I explained that it wasn't a function, but a relationship as it was one to many. I hazarded a guess that you could notate such a relationship as

f(x): R-->R

Where R is the real numbers.

I'm not happy with that as I'm not convinced it does what he is asking, and i'm not convinced my notation is right.

1) Is there a way to notate what he is asking?
2) Is there anything interesting about his question that I can inspire him with? Is this type of thing used anywhere for anything?

As an aside, the same kid had a fantastic question last lesson about what happens to the shape of the graph x^n + y^n = 1 as n->infinity. Unbelievably he answered this question intuitively (for even n), without even reaching for his pencil!
y=f(x) is a one dimensional line so it doesn't "colour in" the full two dimensional x-y plane.

I think you need a 3 dimensional function that defines a 2 dimensional surface like x^2+y^2+z^2=1 defines the surface of a sphere.

PS smart kid!
Post edited at 16:09

For every Jack there's a Jill.

something like y=tan (wx) as w goes to zero would probably do a good job ?

> Is there anything interesting about his question that I can inspire him with?

Yes - a whole fascinating universe. Try googling "space filling curves"

> As an aside, the same kid had a fantastic question last lesson about what happens to the shape of the graph x^n + y^n = 1 as n->infinity. Unbelievably he answered this question intuitively (for even n), without even reaching for his pencil!

That's a lovely question. The odd case is fun too.

> something like y=tan (wx) as w goes to zero would probably do a good job ?

I think you must mean as w tends to infinity, not zero.

And I'm not convinced it does a good job! Maybe......
Post edited at 17:17
In Cartesian coordinates :

z=0

Edit : you could introduce dot & cross product / planar notation as a new concept.
Post edited at 17:36

So, there isn't a single-valued function R -> R as defined, because single valued means it can only take one value at one point.

Space-filling curves are neat, but they map [0, 1] to [0, 1]^2, or otherwise a line to a space.

Another example that's close to what's being asked for are functions that take every value on every interval:
https://en.wikipedia.org/wiki/Conway_base_13_function
That is to say, on every open interval (a, b), no matter how small, it takes every value in R. The construction of this from scratch was the "something to occupy the ones who are too clever for their own good" question on an undergrad example sheet that I did. I didn't get anywhere near it!
In reply to ablackett: What about x+y=x+y? Would that colour the graph in?

Guess it would be an equation of a 3 dimension plane with z=0.

> In Cartesian coordinates :

> z=0

But that is really just the name of the x/y plane and says nothing about any mapping or function.

> But that is really just the name of the x/y plane and says nothing about any mapping or function.

Given that what OP is looking for is a mathematical statement that says "x and y have no relationship" then I suspect that's the only way.

> Given that what OP is looking for is a mathematical statement that says "x and y have no relationship" then I suspect that's the only way.

Or, rather, there isn't a way!

I'm an engineer, not a pure maths guy.

If he wants to plot a graph that's filled-in, he's drawing a plane ( r dot n = c). It's a loci type thing. r=[x,y,z], n=[0,0,1], c=0

If you want strictly define a mapping, I guess there's some set terminology that just describes it, but useless for anything real world. Not a clue I'm afraid.

> I think you must mean as w tends to infinity, not zero.

> And I'm not convinced it does a good job! Maybe......

Infinity, sorry.

> Or, rather, there isn't a way!

Well, kind of.

If he's after a function f(x) such that the graph y = f(x) fills the whole plane then there isn't. The closest you'll get is taking every value on every interval, as above.

If he's after a function f(x, y) such that the set of solutions to f(x, y) = 0 fills the whole plane then f: (x, y) -> 0 does just fine.

All the graph formulae and definitions are constraining what points are valid. Therefore I would say that the empty statement "" would define the entire plane. Tautologies such as x+y=x+y reduce down to be the same thing.
Thank you all, I was hoping to get something interesting out of this and the Space Filling Curves that Robert pointed at are brilliant. We have looked at fractals a bit and how finite lines can have infinite length, and how finite objects can have infinite surface areas so the Peano Curve will be interesting for him to look at.

The Conway base 13 function is a bit trickier for me to get my head around.

Trying to understand the conway base 13 function, if I take the interval (1, 2) and try and find the value in this interval which maps to -8.3 - just picking any old number out of my ass.

If I take the numbe 1.B8C3 (base 13), which is in the interval (1,2)
f(B8C3)=-8.3

> All the graph formulae and definitions are constraining what points are valid. Therefore I would say that the empty statement "" would define the entire plane. Tautologies such as x+y=x+y reduce down to be the same thing.

This might be true, but I don't think it's going to inspire him if I tell him the answer to his question is " ".

I would have liked that answer a lot as a kid, but I am odd! (And ended up doing a lot of maths)

> The Conway base 13 function is a bit trickier for me to get my head around.

> Trying to understand the conway base 13 function, if I take the interval (1, 2) and try and find the value in this interval which maps to -8.3 - just picking any old number out of my ass.

> If I take the numbe 1.B8C3 (base 13), which is in the interval (1,2)

> f(B8C3)=-8.3

Yes, spot on!

Cheers. What an obscure bit of maths.