Part CCSIR NET December 2024the-parts-need-not-have-densities

The parts need not have densities

For a positive integer n and a subset S of the set of positive integers, let S(n) denote the set {s ∈ S | s ≤ n}. Let X be a subset of the set of positive integers such that $lim (n \to \infty)$ |X(n)|/n = 1. Assume that there exist pairwise disjoint subsets $X_{1}, X_{2}, \dots, X_{8}$ of X such that ⋃ $(i = 1..8) X_{i} = X$ . Which of the following statements are true?

A. $lim (n \to \infty)$ | $X_{i} (n)$ |/n exists for all 1 ≤ i ≤ 8.
B. $lim in f (n \to \infty)$ | $X_{i} (n)$ |/n ≥ 0 for all 1 ≤ i ≤ 8.✓
C. $lim sup (n \to \infty)$ | $X_{i} (n)$ |/n ≥ 1/8 for some 1 ≤ i ≤ 8.✓
D. $lim sup (n \to \infty)$ | $X_{i} (n)$ |/n < 1/8 for all 1 ≤ i ≤ 8.

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.

See pricing

50 are analysed free — try those first.

Related counterexample: limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

The parts need not have densities

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