Multivariable function that is injective?

  • I
  • Thread starter NotASmurf
  • Start date
  • #1
NotASmurf
150
2
Hey all, is it possible to find a function that for $$ a,b,c.. \in \mathbb{R} $$ $$ y= f(a,b,c,..) , \hspace{5mm} y= \rho , \rho \in \mathbb{R} \hspace{2mm} for \hspace{2mm} only \hspace{2mm} 1 \hspace{2mm} set \hspace{2mm} of \hspace{2mm} a,b,c.. $$
Any help appreciated
 

Answers and Replies

  • #3
NotASmurf
150
2
I'm fine with any that isn't completely trivial (if there are any trivial solutions), will try adapt to whatever I can get, basically this is for a program that has a graph, and a vertex has to have a single number input as a function of the labels of the vertices already in the path. Inputing the entire path so far will take up way too many resources.
 
  • #4
36,025
12,921
There are injective functions ##g: \mathbb{R} \times \mathbb{R} \to \mathbb{R}##, you can use f(a,b,c)=g(a,g(b,c)). Those functions are messy, and need infinite precision to be truly injective, I'm quite sure you don't want to use them.

More context would help, but I guess there is an easier solution. Why can't you just use the set of three numbers as label? Expressed as string or whatever if the data format is an issue.
 
  • #5
NotASmurf
150
2
Don't want to pass entire label as worsens the time complexity of the program by at the minimum of increasing the power by 1. /: Could you please list an example of one of the functions which satisfies $$ \mathbb{R} \times \mathbb{R} \to \mathbb{R} $$ ?
 
  • #6
wrobel
Science Advisor
Insights Author
997
858
The sets ##\mathbb R^m## and ##\mathbb R^n## have the same cardinality for all ##m,n##
 
  • #7
36,025
12,921
Here is an example
Your label will have to be longer than the length of a single coordinate, in a suitable format. If you have N possible values for the single coordinate, you need N3 possible labels.

I don't see how concatenating strings would increase the time complexity of anything.
 
  • #8
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,302
Maybe you're interested in a function ##\mathbb{Q}\times \mathbb{Q}\rightarrow \mathbb{Q}## instead? That is much easier to give, but I'm sure it's not going to be useful in a computational context.

In any case, given ##m/n## and ##m'/n'## in reduced form (meaning that ##m## and ##n## have no common divisors and ##n>0## and likewise for ##m'## and ##n'##), you can send this to ##2^m 3^n 5^{m'} 7^{n'}##.
 
  • #9
NotASmurf
150
2
I don't see how concatenating strings would increase the time complexity of anything.

The processing required for what those strings will be processed as it will
 
  • #10
NotASmurf
150
2
Maybe you're interested in a function ##\mathbb{Q}\times \mathbb{Q}\rightarrow \mathbb{Q}## instead? That is much easier to give, but I'm sure it's not going to be useful in a computational context.

In any case, given ##m/n## and ##m'/n'## in reduced form (meaning that ##m## and ##n## have no common divisors and ##n>0## and likewise for ##m'## and ##n'##), you can send this to ##2^m 3^n 5^{m'} 7^{n'}##.
What's $$ m' $$ and $$n'$$?
 
  • #11
36,025
12,921
The processing required for what those strings will be processed as it will
It is linear in the number of vertices, which is as good as it can get.

m' and n' are the numerator and denominator of the second fraction.
 

Suggested for: Multivariable function that is injective?

  • Last Post
Replies
9
Views
15K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
12
Views
8K
  • Last Post
Replies
32
Views
7K
  • Last Post
Replies
6
Views
800
  • Last Post
Replies
2
Views
10K
Replies
3
Views
7K
Replies
3
Views
944
Replies
9
Views
6K
  • Last Post
Replies
6
Views
1K
Top