Set of all rational sequences countable?

In summary, the conversation discusses the countability of rational numbers and their cartesian products. The speaker mentions that Q, the set of rational numbers, and Q^n, the finite cartesian product of Q with itself, are both countable. They then question whether the countably infinite cartesian product of Q with itself is also countable. The speaker also brings up the concept of Q^∞, which is the countable cartesian product of Q with itself. They state that if Q^∞ is countable, then the set of all rational sequences is also countable. The conversation ends with the speaker realizing their initial mistake and thanking the other person for pointing it out.
  • #1
rat bass
4
0
I know that Q (rational numbers) are countable and that the finite cartesian of Q with itself, Q^n is countable but is it true that the countably infinite cartesian product of Q with itself is countable? The set of all rational sequences are isomorphic to Q^∞ (here I am saying Q^∞ is the countable cartesian product of Q with itself) so if I know Q^∞ is countable then I know the set of all rational sequences is countable. I need this result to prove something else for a homework problem and I want to be able to justify it.
 
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  • #2
Think about the real numbers.
 
  • #3
ah nevermind I see it. Thank you I was wrong.
 

1. What is a "set of all rational sequences"?

A set of all rational sequences is a collection of infinite sequences of rational numbers, where each term in the sequence is a rational number. A rational number is any number that can be expressed as a ratio of two integers.

2. Why is the set of all rational sequences countable?

The set of all rational sequences is countable because it can be put into a one-to-one correspondence with the set of natural numbers, which is a countably infinite set. This means that each rational sequence can be identified with a unique natural number, and vice versa.

3. How is the countability of the set of all rational sequences proven?

The countability of the set of all rational sequences can be proven using Cantor's diagonal argument. This involves constructing a list of all possible rational sequences and then showing that there is always a sequence that is not included in the list. This contradiction proves that the set is uncountable.

4. What are some examples of rational sequences?

Some examples of rational sequences include:
- The sequence 1, 1/2, 1/3, 1/4, ...
- The sequence 0, 1/2, 2/3, 3/4, ...
- The sequence 1/2, 2/3, 3/4, 4/5, ...

5. How is the set of all rational sequences useful in mathematics?

The set of all rational sequences is useful in mathematics because it helps to define and understand important concepts such as limits, convergence, and continuity. It also plays a crucial role in number theory and analysis. Additionally, the set of all rational sequences is often used as a stepping stone to understand more complex sets, such as the set of real numbers.

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