Set of sequences of rationals.

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In summary, the conversation revolves around proving that there are c sequences of rational numbers, and specifically showing that |Q^N| = c. Several attempts are made, including considering a bijection from {0,1}^N to Q^N and using the fact that any real number can be identified with a specific sequence of rational numbers. The axiom of choice is mentioned as potentially being necessary for the proof. The possibility of using the fact that |Q| = |N| is also suggested.
  • #1
MathematicalPhysicist
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i need to prove that there are c sqequences of rational numbers.
basically, i need to show that |Q^N|=c.
here, are a few attempts from my behalf:
i thought that Q^N is a subset of R^N, so |Q^N|<=c, but this doesn't help here, so i thought perhaps to find a bijection from {0,1}^N to Q^N.
i know that each rational number can be represented in base 2 by the digits 0,1, but I am having difficulty to formalise this idea.
 
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  • #2
Any real number can be identified with a specific sequence of rational numbers. For example, the real number [itex]\pi[/itex] can be identified with the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, with each term (a rational number because it is a terminating decimal) being one more decimal place in the decimal expansion of [itex]\pi[/itex]. This gives a one-to-one function from the set set of real numbers to the set of all sequences of rational numbers and so shows that [itex]c\le |Q^N|[/itex].
 
  • #3
Grr. If I remember correctly, it's a theorem that for infinite c:

[tex]a^c = b^c[/tex]

whenever [itex]a \leq b \leq c[/itex]. The proof isn't immediately leaping to mind, though. :frown:

Oh, bother, I just checked Wikipedia, and it looks like you need the axiom of choice to prove that, so the proof won't be as easy as I had hoped.

Anyways, LQG, remember that |Q| = |N|. It might be easier to try and prove |N||N| = 2|N| or |N||N| = |R| instead.

But, we see HoI seems to have proven it directly, so don't listen to me. :wink:
 
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  • #4
Oh, Hurkyl, I'm blushing.


By the way, I had a friend who, on another forum, used the name "Hog on Ice" which was regularly abbreviated "HOI". So whenever I see "HOI" used for "HallsofIvy", I'm surprized!
 

1. What is a set of sequences of rationals?

A set of sequences of rationals is a collection of infinite sequences of rational numbers. These sequences can have a pattern or be randomly generated, but they must consist of only rational numbers.

2. How is a set of sequences of rationals different from a set of rational numbers?

A set of sequences of rationals is different from a set of rational numbers in that it contains infinite sequences of rational numbers, rather than just individual rational numbers.

3. How are sequences of rationals useful in scientific research?

Sequences of rationals are useful in scientific research because they can be used to model real-life processes that involve continuous change, such as the growth of bacteria or the trajectory of a projectile. They can also be used to represent data that is collected over time.

4. Can a sequence of rationals have an infinite number of unique values?

Yes, a sequence of rationals can have an infinite number of unique values. For example, the sequence 1/2, 1/4, 1/8, 1/16,... will have an infinite number of unique values as it continues towards infinity.

5. What is the significance of a sequence of rationals converging to a limit?

When a sequence of rationals converges to a limit, it means that the values in the sequence are getting closer and closer to a specific value as the sequence progresses. This can be useful in scientific research for predicting future values or understanding patterns in data.

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