Increasing bijection
Does there exist a strictly increasing bijective function f which maps from the rationals to the rationals without 0?
In symbols that is f : Q → Q ∖ { 0 } .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Brilliant solution! The back-and-fourth method is such a powerful tool!
If anyone is curious, I will give an example of such a function f .
We use the digits of π = 3 . 1 4 1 5 9 . . .
f ( 3 ) = − 3 1 , f ( 3 . 1 ) = − 3 1 1 , f ( 3 . 1 4 ) = − 3 1 4 1 and so on...
f ( 4 ) = 4 1 , f ( 3 . 2 ) = 3 2 1 , f ( 3 . 1 5 ) = 3 1 5 1 and so on...
We then do linear interpolation between the plotted points and for the end points we plot a line of gradient 1 which extends to infinity at each side.
If this is unclear (which it kind of is without a graph to point at and explain with) then please ask.
This certainly isn't the most efficient example of such a function, it's the first explicit example I thought of.
Log in to reply
Yeah, actually creating such a function is the tricky part. :) For your first year you sure are covering a lot of material, (assuming that your posted questions are the result of material you're covering in your lectures/tutorials). I guess with a 3-year programme you have to go at an accelerated rate; in my first year it was just wall-to-wall calculus and linear algebra. The fun stuff like number theory, topology and foundations of analysis didn't start until second year.
Log in to reply
A few of my questions have been inspired by what I have studied, most of them haven't I would say. I do plan on posting problems related to the things I'm learning soon though.
The things I've learnt so far are:
- Number and set theory.
- Basic linear algebra.
- Differential equations.
- Basic group theory.
The things I'm learning now are:
- Analysis.
- Dynamics and relativity.
- Vector calculus.
- Probability.
- Advanced group, ring and module theory.
- Complex methods.
- Electromagnetism.
(and more in my spare time but I don't take them officially, just for fun).
5,6 and 7 are second year topics. I'm doing them now so when I'm in my second year I will be able to do all of the second year mathematics. There is too much choice that I go to extra lectures in order to learn as much as I can... but it's impossible to learn it all in 3 years...
How long was your whole course?
Log in to reply
@Isaac Buckley – Wow, I'm impressed by the variety of topics that you are covering/have covered in your first year. I think that the 'system' there is more flexible than in most North American universities, allowing you to go to extra lectures out of a desire to simply "learn as much as you can". I laughed when I read "... and more in my spare time". I greatly admire your enthusiasm; it far exceeds that which I ever possessed. :)
My programme took 5 years, as I took a co-op degree with a double honours in mathematics and physics. (The co-op part involved alternating work and academic terms, with the work terms being physics-related.) In first year I also took chemistry, computer science and English literature before focusing exclusively on physics and mathematics from second year on. However, you'll probably have covered more material after two years than I did in my entire programme. Do you also have to do some kind of undergraduate thesis to complete your degree?
Since both Q and Q ∖ { 0 } are countably infinite, densely ordered sets without endpoints, they are isomorphic , (proof via the back-and-forth method ), and an isomorphism between such sets is a strictly increasing bijection.