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GATE 2023 ST – Question 49

Calculus · Sequences and Series · 2 marks · Multiple select

Which sets are countable?

  1. All functions from {1,…,10} to ℚ.
  2. All functions from ℕ to {0,1}.
  3. All integer-valued sequences with finitely many nonzero terms.
  4. All integer-valued sequences converging to 1.

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Show answer and explanation

Correct answer: (A) All functions from {1,…,10} to ℚ.; (C) All integer-valued sequences with finitely many nonzero terms.; (D) All integer-valued sequences converging to 1.

Explanation

A is ℚ¹⁰,a finite product of countable sets. B is uncountable by the binary-sequence diagonal argument. C is a countable union over finite support sizes and positions of finite integer tuples. An integer sequence converging to 1 is eventually identically 1,so D is also a countable union of finite integer prefixes. **A,C,D**.

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