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Part CCSIR NET December 2023partition-counterexamples

Partition counterexamples

Let {AnA_{n}}n≥1_{n\ge1} be a collection of non-empty subsets of Z\mathbb{Z} such that An∩Am=∅A_{n} \cap A_{m} = \emptyset for m ≠ n. If Z=∪n≥1\mathbb{Z} = \cup_{n\ge1} AnA_{n}, then which of the following statements are necessarily true?

  1. A.AnA_{n} is finite for every integer n ≥ 1.
  2. B.AnA_{n} is finite for some integer n ≥ 1.
  3. C.AnA_{n} is infinite for some integer n ≥ 1.
  4. D.AnA_{n} is countable (finite or infinite) for every integer n ≥ 1.✓ Correct

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: A subset of ℝ with measure zero is countable

More on this topic

The chapter behind this: Lebesgue measure and measurable functions — free to read

From Lebesgue Measure and Integration › Measurable sets and functions

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