Part BCSIR NET December 2025restricting-digits-to-a-fixed-alphabet-is-a-cantor-set-argument-not-a-countability-argument

Restricting digits to a fixed alphabet is a cantor set argument not a countability argument

Let S be a subset of the open interval (0,1) that consists of all the real numbers $\alpha \in (0,1)$ whose infinite decimal expansion $\alpha = 0.a_{1}a_{2}a_{3}\cdots$ is such that $a_{i} \in$ {0,2,4} for all i ≥ 1. Which of the following statements is true?

A.There is a bijective map from $\mathbb{N}$ to S.
B.There is a surjective map from S onto (0,1).✓ Correct
C.There is a bijective map from $\mathbb{N}$ to $(0,1)\S$ .
D.S is a countable set and $(0,1)\S$ is uncountable.

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.

See pricing

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.

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Restricting digits to a fixed alphabet is a cantor set argument not a countability argument

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