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Part CCSIR NET December 2024only-an-interval-condition-leaves-uncountably-many

Only an interval condition leaves uncountably many

For a real number x, let [x] denote the largest integer ≤ x. Which of the following sets are uncountable?

  1. A.{x∈Rx \in \mathbb{R} | [x] = 1}✓ Correct
  2. B.{x∈Rx \in \mathbb{R} | x − [x] = 1/2}
  3. C.{x∈Rx \in \mathbb{R} | x≥0,x∈Qx \ge 0, \sqrt{x} \in \mathbb{Q}}
  4. D.{x∈Rx \in \mathbb{R} | x2∈Qx^{2} \in \mathbb{Q}}

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Every ordered field is Archimedean

More on this topic

The chapter behind this: The real line: supremum, Archimedes and density — free to read

From The Real Line › Completeness, sup/inf, Archimedean property

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