Exercise · ZW-CONSTRUCTION-OF-THE-REALS-010 · ★★★ Advanced
Let \( A=\{q: q<0 \text{ or } q^2<2\} \) and \( B=\{q: q<0 \text{ or } q^2<2\} \). What is \( A\cdot B \) as a real number, in the positive case?
Choose one option.
Show answer
- B. \( 2 \)
Explanation
Each cut represents \( \sqrt{2} \), and the product of positive cuts corresponds to the product of their values, \( \sqrt{2}\cdot\sqrt{2}=2 \).
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-001 Which of the following sets is a Dedekind cut?
- 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 ?