Cardnality of Infinite Sets

In summary: I'm saying that c^c is the cardinal number of things that are DETERMINED by their values on the rationals Q. Q is countable.
  • #1
kingwinner
1,270
0
1) Find the cardnality of the set of continuous functions from R to R.

Let's consider a simplified version: for the cardnality of the set of functions from R to R, I can compare it with the set of characteristic functions of subsets of R and conclude that they both have cardnality 2c

But when the word "continuous" is inserted, how can I find its cardnality?

Thanks for any help!
 
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  • #2
If f is a continuous function and you know the value of f(q) for all rational q, do you know the value of f for any real value?
 
  • #3
Dick said:
If f is a continuous function and you know the value of f(q) for all rational q, do you know the value of f for any real value?

I think it would be hard to tell specifically. Since f is continuous, the values can't go too far, but there is still a wide possible range. Another thing is that we may not know the value of f(q) for q rational
 
  • #4
Also,
since |{functions from R to R}|=2c,
I think that |{continuous functions from R to R}|<2c.
This narrows down the possible answers, but I still don't know how to get the cardnality exactly.
 
  • #5
kingwinner said:
I think it would be hard to tell specifically. Since f is continuous, the values can't go too far, but there is still a wide possible range. Another thing is that we may not know the value of f(q) for q rational

That was a rhetorical question. Let me put it this way, if you know the value of f for all rationals, then you know the value of f for all reals. Why do I believe this? That means while a general functions have cardinality c^c the cardinality of continuous functions may be less. Is it?
 
  • #6
Dick said:
That was a rhetorical question. Let me put it this way, if you know the value of f for all rationals, then you know the value of f for all reals. Why do I believe this? That means while a general functions have cardinality c^c the cardinality of continuous functions may be less. Is it?

Sorry, I am lost for two reasons...

The question asks for the cardnality of the set of continuous functions from R to R, so shouldn't we not be assuming any further detail. It didn't say that the value of f for all rationals are known. But perhaps I am misunderstanding the question...

Also, why is the answer c^c? I haven't encountered the cardinal number c^c so I don't know what it means. Shoudn't the answer be less than or equal to 2^c?
 
  • #7
c^c=2^c. Wasn't that what you showed in the first part? What I'm saying is that continuous functions are DETERMINED by their values on the rationals Q. Q is countable.
 
  • #8
kingwinner said:
Sorry, I am lost for two reasons...

The question asks for the cardnality of the set of continuous functions from R to R, so shouldn't we not be assuming any further detail. It didn't say that the value of f for all rationals are known. But perhaps I am misunderstanding the question...
That's not the point. Let x be any real number. Let {xn} be a sequence converging to x. Since f is continuous at x, [itex]\lim_{n\rightarrow \infty} f(x_n)= f(x)[/itex]. That is, if you two continuous functions are equal on all rational numbers, then they are equal on all real numbers. Therefore the cardinality of the set of all continuous functions from R to R cannot be larger than the cardinality of the set of all functions from Q to R.

Also, why is the answer c^c? I haven't encountered the cardinal number c^c so I don't know what it means. Shoudn't the answer be less than or equal to 2^c?
Where did you see that the answer was cc?
 

1. What is the definition of cardnality of infinite sets?

The cardnality of infinite sets refers to the number of elements or items in a set that is infinite, meaning it has no specific end or limit. It is represented by the symbol ∞ and is a concept used in mathematics to compare the sizes of different infinite sets.

2. How is the cardnality of infinite sets determined?

The cardnality of infinite sets is determined by counting the number of elements in the set and comparing it to the number of elements in another set. If both sets have the same number of elements, then they have the same cardnality. If one set has more elements than the other, then it has a higher cardnality.

3. Are all infinite sets of the same cardnality?

No, not all infinite sets have the same cardnality. There are different levels or sizes of infinite sets, and they can be compared using mathematical techniques such as bijections and injections. For example, the set of all real numbers (represented by ℝ) has a higher cardnality than the set of all natural numbers (represented by ℕ).

4. Can the cardnality of infinite sets change?

No, the cardnality of infinite sets cannot change. It is a fixed value that represents the size of the set. However, the cardnality of a set can be equivalent to the cardnality of another set, meaning they have the same number of elements even if the elements themselves are different.

5. How does the concept of cardnality of infinite sets apply in real life?

The concept of cardnality of infinite sets is used in various fields such as computer science, physics, and economics. In computer science, it is used in data structures and algorithms. In physics, it is used in concepts such as infinity and the measurement of the universe. In economics, it is used in the study of infinite goods and services.

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