Partially Ordered Sets · Partial Orders

Lesson 2

Nikolai Chukhin · Alexander S. Kulikov

Problem. Mark all sets from which it is possible to choose a decreasing sequence of infinite length (that is, a sequence \((x_{1}, x_{2}, x_{3}, \dotsc)\), for which \(x_{1}>x_{2}>x_{3}> \dotsb\)).

5 points

  1. \(\mathbb{Z}_{\ge 0}\) (non-negative integers)

  2. \(\mathbb{Z}\) (integers)

  3. \(\mathbb{P}\) (primes; recall that they are positive by definition)

  4. \(\mathbb{Q}\) (rationals)

  5. \(\mathbb{R}\) (reals)

  6. \([2, 3]\) (\([2, 3]=\{x \in \mathbb{R}\colon 2 \le x \le 3\}\))