Part CCSIR NET December 2023partition-counterexamples

Partition counterexamples

Let { $A_{n}$ } $_{n \geq 1}$ be a collection of non-empty subsets of $Z$ such that $A_{n} \cap A_{m} = \emptyset$ for m ≠ n. If $Z = \cup_{n \geq 1}$ $A_{n}$ , then which of the following statements are necessarily true?

A. $A_{n}$ is finite for every integer n ≥ 1.
B. $A_{n}$ is finite for some integer n ≥ 1.
C. $A_{n}$ is infinite for some integer n ≥ 1.
D. $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.

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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

Partition counterexamples

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