Is the set difference ℝ \ R still uncountable

discrete mathematicselementary-set-theory

I know that the set of real numbers are uncountable by cantors diagonal argument.
But the set difference of ℝ and R (Real numbers minus itself) must equal an empty set right? And an empty set is itself countable? Or is it a trick question where they are only defining R as a subset (not all) of real numbers?

Question: Consider the set R of real numbers which can be written in one of the usual ways: strings of digits with an optional negative sign and/or an optional decimal point, a fraction sign between such strings, and/or such strings with square root signs.

  • Is this set R finite; infinite but countable; or not countable? Explain your answer

  • Is ℝ \ R countable or uncountable?

Best Answer

I believe that $\mathbb{R} \setminus R$ is uncountably infinite. The set $R$ contains $\mathbb{Q}$, and is contained inside the set of real algebraic numbers. But the set of real algebraic numbers is countably infinite. Hence,

  1. $R$ is countably infinite.
  2. $\mathbb{R} \setminus R$ is uncountably infinite.

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