Set of all rational sequences countable?

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SUMMARY

The discussion confirms that the set of all rational sequences, represented as Q^∞ (the countable Cartesian product of rational numbers Q with itself), is indeed countable. This conclusion is based on the established fact that both Q and any finite Cartesian product of Q, denoted as Q^n, are countable. The participant initially sought to justify this result for a homework problem but later acknowledged a misunderstanding regarding the countability of Q^∞.

PREREQUISITES
  • Understanding of countable sets in set theory
  • Familiarity with Cartesian products of sets
  • Knowledge of rational numbers (Q)
  • Basic concepts of isomorphism in mathematics
NEXT STEPS
  • Study the properties of countable and uncountable sets in set theory
  • Explore the concept of Cartesian products in more depth
  • Investigate the implications of isomorphism in set theory
  • Review proofs related to the countability of infinite sets
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Mathematicians, students studying set theory, and anyone interested in the properties of rational numbers and their sequences.

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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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Think about the real numbers.
 
ah nevermind I see it. Thank you I was wrong.
 

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