Exercise · ZW-CONSTRUCTION-OF-THE-REALS-001 · ★☆☆ Basic
Which of the following sets is a Dedekind cut?
Choose one option.
Show answer
- C. \( \{q\in\mathbb{Q}: q<1\} \)
Explanation
The rationals less than \( 1 \) form a nonempty set, closed downward, with no maximum. The empty set is excluded, \( \{q>1\} \) is not closed downward, and \( \{q\le 1\} \) has a maximum.
Review the concept: Construction of the reals
The real numbers \( \mathbb{R} \) form a complete ordered field extending \( \mathbb{Q} \), in which every bounded Cauchy sequence converges. They are built from \( \mathbb{Q} \) using Dedekind cuts or Cauchy sequences.
\[\mathbb{R}=\{A\subseteq\mathbb{Q}: A \text{ es cortadura}\}\]
More practice problems: construction of the reals
- ZW-CONSTRUCTION-OF-THE-REALS-002 What property must a set A have to be a cut?
- ZW-CONSTRUCTION-OF-THE-REALS-003 Which real number does the cut of negative rationals or rationals with square less than 4 represent?
- ZW-CONSTRUCTION-OF-THE-REALS-004 Which set of rationals defines the cut of √3 ?
- ZW-CONSTRUCTION-OF-THE-REALS-005 A Dedekind cut A must differ from ℚ . Why?
- ZW-CONSTRUCTION-OF-THE-REALS-006 Let A=\q∈ℚ: q<0 or q^2<5\ . Which real does A represent?
- ZW-CONSTRUCTION-OF-THE-REALS-007 What is the supremum in ℝ of the cut A=\q∈ℚ: q<1/2\ ?
- ZW-CONSTRUCTION-OF-THE-REALS-008 If A and B are cuts, what is the cut of A+B in terms of A and B ?
- ZW-CONSTRUCTION-OF-THE-REALS-009 Why is ℚ not complete while ℝ is?