Question

Какова мощность множества всех рациональных функций с целыми коэффициентами в числителе и знаменателе?

Answer 1

Мощность множества всех рациональных функций с целыми коэффициентами в числителе и знаменателе не более, чем счетно. Так как мощность рациональных чисел тоже не более, чем счетно.

Answer 2

Rational Functions with Integer Coefficients

The power of the set of all rational functions with integer coefficients in the numerator and denominator is uncountably infinite. This means that the cardinality of this set is greater than that of the set of natural numbers, integers, rational numbers, and even real numbers.

This can be understood through the concept of cardinality in set theory. The set of rational functions with integer coefficients is so vast that it cannot be put into a one-to-one correspondence with the set of natural numbers, which is the definition of countable sets.

In essence, the set of all rational functions with integer coefficients is incredibly large and cannot be enumerated in a systematic way like the set of natural numbers or even the set of real numbers.

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