Cardinality of Functions Mapping Irrational Numbers to Rational Numbers

[trap101]

Homework Statement

Let S be the set of all functions mapping the set {√2, √3, √5,√7} into Q. What is the cardinality of S?

Homework Equations

The Attempt at a Solution

I have been stuck staring and trying to think of something to figure out this question. This is the idea i have:

let U = { all irrational numbers} V = {√2, √3, √5,√7}. Define a set T = {g: U--> Q} Now I think it might be easier for me to get the cardinality of T and since S would be a subset of that then perhaps go from there?

Now looking at it again I don't think it's as swell of an idea. But it's the only thing I got to try. With that, I'm still stuck in tyring to associate T with a cardinality I know (in this case I think it's going to be associated to the cardinality of Q).

Do I have to come up with a specific bijective function that would work?

 

[mfb]

The problem statement is ambiguous:
* Do we have functions f: R->R, with the constraint that $f(\sqrt{2})$, ... are rational?
* Do we have functions F: {√2, √3, √5,√7} -> Q?
* Something else?

In the first case, do you know the cardinality of all functions f:R->R?
In the second case, it does not matter that you have 4 irrational numbers. It is just a set with 4 elements.

 

[trap101]

The problem statement is ambiguous:
* Do we have functions f: R->R, with the constraint that $f(\sqrt{2})$, ... are rational?
* Do we have functions F: {√2, √3, √5,√7} -> Q?
* Something else?

In the first case, do you know the cardinality of all functions f:R->R?
In the second case, it does not matter that you have 4 irrational numbers. It is just a set with 4 elements.

The closest thing I have in the first case is that I have proved the cardinality of the set of functions that map from N --> R. {f: N-->R }. There have been no constraints established.

In terms of the second case: There are no specific functions stated that F: {√2, √3, √5,√7} -> Q.


In thinking about the second case, I was wondering if I should be only considering the elements one at a time or can I combine the elements? I ask because I was thinking if I could create a function that could go from my set to the rationals. It would be easy if I only had to consider each element individually. But I don't think that is the case.

 

[vela]

mfb was asking for clarification on what exactly the problem statement was because your approach seemed unnecessarily complicated if the problem statement was taken at face value. He wasn't asking you to discuss each case.

Suppose instead that the problem was asking you to consider functions from {1, 2, 3, 4} into Q. Would that change the cardinality of S?

 

[trap101]

mfb was asking for clarification on what exactly the problem statement was because your approach seemed unnecessarily complicated if the problem statement was taken at face value. He wasn't asking you to discuss each case.

Suppose instead that the problem was asking you to consider functions from {1, 2, 3, 4} into Q. Would that change the cardinality of S?

No it wouldn't. But I still have to some how get the cardinality of that set. I suppose by comparing it to another set because there are an infinite amount of functions that could create that mapping.

 

[vela]

If the type of object in the domain of the functions doesn't matter, why were you asking about the cardinality of the irrationals?

 

[mfb]

It is still unclear to me what exactly you are supposed to do. Based on the given problem statement, I would expect functions F: {√2, √3, √5,√7} -> Q, but then it is strange that those four roots were chosen for the problem.

If you did not give the exact problem statement in the first post, please post it here exactly as you got it.